import ECTate.Algebra.EllipticCurve.Kronecker
import ECTate.Algebra.EllipticCurve.Model
import ECTate.Algebra.ValuedRing
import ECTate.Data.Nat.Enat
import Mathlib.Data.Int.Basic
import Mathlib.Tactic.LibrarySearch
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Contrapose

open Enat

variable {
R: Type u
R : 
Type u: Type (u + 1)
Type u} [
inst: IntegralDomain R
inst : 
IntegralDomain: Type ?u.19993 → Type ?u.19993
IntegralDomain 
R: Type u
R]


namespace Model

variable {
p: R
p : 
R: Type u
R}

def 
is_local_singular_point: {R : Type u} → [inst : IntegralDomain R] → {p : R} → SurjVal p → Model R → R × R → Prop
is_local_singular_point (
valp: SurjVal p
valp : 
SurjVal: {R : Type ?u.21} → R → [inst : IntegralDomain R] → Type ?u.21
SurjVal 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.30) → [inst : IntegralDomain R] → Type ?u.30
Model 
R: Type u
R) (
P: R × R
P : 
R: Type u
R × 
R: Type u
R) : 
Prop: Type
Prop :=
valp: SurjVal p
valp.
v: {R : Type ?u.43} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v (
weierstrass: {R : Type ?u.50} → [inst : IntegralDomain R] → Model R → R × R → R
weierstrass 
e: Model R
e 
P: R × R
P) > 
0: ?m.58
0 ∧ 
valp: SurjVal p
valp.
v: {R : Type ?u.312} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v (
dweierstrass_dx: {R : Type ?u.316} → [inst : IntegralDomain R] → Model R → R × R → R
dweierstrass_dx 
e: Model R
e 
P: R × R
P) > 
0: ?m.320
0 ∧ 
valp: SurjVal p
valp.
v: {R : Type ?u.338} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v (
dweierstrass_dy: {R : Type ?u.342} → [inst : IntegralDomain R] → Model R → R × R → R
dweierstrass_dy 
e: Model R
e 
P: R × R
P) > 
0: ?m.346
0

lemma 
singular_of_val_discr: ∀ (valp : SurjVal p) (e : Model R), SurjVal.v valp (discr e) > 0 → ∃ P, is_local_singular_point valp e P
singular_of_val_discr
Warning: declaration uses 'sorry' (
valp: SurjVal p
valp : 
SurjVal: {R : Type ?u.383} → R → [inst : IntegralDomain R] → Type ?u.383
SurjVal 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.392) → [inst : IntegralDomain R] → Type ?u.392
Model 
R: Type u
R) (
h: SurjVal.v valp (discr e) > 0
h : 
valp: SurjVal p
valp.
v: {R : Type ?u.397} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
discr: {R : Type ?u.404} → [inst : IntegralDomain R] → Model R → R
discr > 
0: ?m.413
0) :
  ∃ 
P: ?m.671
P, 
is_local_singular_point: {R : Type ?u.673} → [inst : IntegralDomain R] → {p : R} → SurjVal p → Model R → R × R → Prop
is_local_singular_point 
valp: SurjVal p
valp 
e: Model R
e 
P: ?m.671
P :=
by
  
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: Model R

h: SurjVal.v valp (discr e) > 0

∃ P, is_local_singular_point valp e Psorry
Goals accomplished! 🐙

def 
move_singular_point_to_origin_triple: {R : Type u} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → R × R × R
move_singular_point_to_origin_triple (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.707} → R → [inst : IntegralDomain R] → Type ?u.707
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.716) → [inst : IntegralDomain R] → Type ?u.716
Model 
R: Type u
R) : 
R: Type u
R × 
R: Type u
R × 
R: Type u
R :=
  match 
evr: EnatValRing p
evr.
residue_char: {R : Type ?u.734} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → ℕ
residue_char with
  | 
2: ℕ
2 => (
evr: EnatValRing p
evr.
norm_repr: {R : Type ?u.758} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → R → R
norm_repr 
e: Model R
e.
a4: {R : Type ?u.762} → [inst : IntegralDomain R] → Model R → R
a4, 
0: ?m.772
0, 
evr: EnatValRing p
evr.
norm_repr: {R : Type ?u.835} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → R → R
norm_repr (
e: Model R
e.
a6: {R : Type ?u.842} → [inst : IntegralDomain R] → Model R → R
a6 + 
e: Model R
e.
a4: {R : Type ?u.848} → [inst : IntegralDomain R] → Model R → R
a4 * 
e: Model R
e.
a2: {R : Type ?u.851} → [inst : IntegralDomain R] → Model R → R
a2))
  | 
3: ℕ
3 => (
evr: EnatValRing p
evr.
norm_repr: {R : Type ?u.1074} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → R → R
norm_repr (-
e: Model R
e.
b6: {R : Type ?u.1079} → [inst : IntegralDomain R] → Model R → R
b6), 
0: ?m.1118
0, 
evr: EnatValRing p
evr.
norm_repr: {R : Type ?u.1120} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → R → R
norm_repr (
e: Model R
e.
a3: {R : Type ?u.1127} → [inst : IntegralDomain R] → Model R → R
a3 - 
e: Model R
e.
b6: {R : Type ?u.1133} → [inst : IntegralDomain R] → Model R → R
b6 * 
e: Model R
e.
a1: {R : Type ?u.1139} → [inst : IntegralDomain R] → Model R → R
a1))
  | _ => (
0: ?m.1279
0, 
0: ?m.1286
0, 
0: ?m.1289
0) --need to fill here

def 
move_singular_point_to_origin_iso: {R : Type u} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
move_singular_point_to_origin_iso (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.2056} → R → [inst : IntegralDomain R] → Type ?u.2056
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.2065) → [inst : IntegralDomain R] → Type ?u.2065
Model 
R: Type u
R) : 
Model: (R : Type ?u.2069) → [inst : IntegralDomain R] → Type ?u.2069
Model 
R: Type u
R :=
rst_triple: {R : Type ?u.2074} → [inst : IntegralDomain R] → Model R → R × R × R → Model R
rst_triple 
e: Model R
e (
move_singular_point_to_origin_triple: {R : Type ?u.2079} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → R × R × R
move_singular_point_to_origin_triple 
evr: EnatValRing p
evr 
e: Model R
e)

lemma 
move_singular_point_to_origin: ∀ {R : Type u} [inst : IntegralDomain R] {p p_1 : R} {valp : SurjVal p_1} (evr : EnatValRing p) (e : Model R),
  (∃ P, is_local_singular_point valp e P) →
    is_local_singular_point valp (move_singular_point_to_origin_iso evr e) (0, 0)
move_singular_point_to_origin
Warning: declaration uses 'sorry' (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.2166} → R → [inst : IntegralDomain R] → Type ?u.2166
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.2171) → [inst : IntegralDomain R] → Type ?u.2171
Model 
R: Type u
R) :
(∃ 
P: ?m.2179
P, 
is_local_singular_point: {R : Type ?u.2181} → [inst : IntegralDomain R] → {p : R} → SurjVal p → Model R → R × R → Prop
is_local_singular_point 
valp: ?m.2163
valp 
e: Model R
e 
P: ?m.2179
P) →
  
is_local_singular_point: {R : Type ?u.2200} → [inst : IntegralDomain R] → {p : R} → SurjVal p → Model R → R × R → Prop
is_local_singular_point 
valp: ?m.2163
valp (
move_singular_point_to_origin_iso: {R : Type ?u.2210} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
move_singular_point_to_origin_iso 
evr: EnatValRing p
evr 
e: Model R
e) (
0: ?m.2222
0, 
0: ?m.2286
0) := by 
R: Type u

inst: IntegralDomain R

p, p✝: R

valp: SurjVal p✝

evr: EnatValRing p

e: Model R

(∃ P, is_local_singular_point valp e P) → is_local_singular_point valp (move_singular_point_to_origin_iso evr e) (0, 0)sorry
Goals accomplished! 🐙

def 
pi_scaling: {R : Type u} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.2324} → R → [inst : IntegralDomain R] → Type ?u.2324
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.2333) → [inst : IntegralDomain R] → Type ?u.2333
Model 
R: Type u
R) : 
Model: (R : Type ?u.2337) → [inst : IntegralDomain R] → Type ?u.2337
Model 
R: Type u
R := {
  a1 := 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.2347} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a1: {R : Type ?u.2360} → [inst : IntegralDomain R] → Model R → R
a1 
1: ?m.2364
1,
  a2 := 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.2374} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a2: {R : Type ?u.2384} → [inst : IntegralDomain R] → Model R → R
a2 
2: ?m.2388
2,
  a3 := 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.2393} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a3: {R : Type ?u.2403} → [inst : IntegralDomain R] → Model R → R
a3 
3: ?m.2407
3,
  a4 := 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.2412} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a4: {R : Type ?u.2422} → [inst : IntegralDomain R] → Model R → R
a4 
4: ?m.2426
4,
  a6 := 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.2431} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a6: {R : Type ?u.2441} → [inst : IntegralDomain R] → Model R → R
a6 
6: ?m.2445
6 }

lemma 
pi_scaling_of_b2: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      EnatValRing.sub_val evr (b2 e) 2 =         EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2
pi_scaling_of_b2 (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.2530} → R → [inst : IntegralDomain R] → Type ?u.2530
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.2539) → [inst : IntegralDomain R] → Type ?u.2539
Model 
R: Type u
R) (
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.2544} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.2551} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a1: {R : Type ?u.2558} → [inst : IntegralDomain R] → Model R → R
a1 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.2564
1)
  (
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.2582} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.2586} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a2: {R : Type ?u.2590} → [inst : IntegralDomain R] → Model R → R
a2 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
2: ?m.2594
2) :
  
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.2604} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
b2: {R : Type ?u.2614} → [inst : IntegralDomain R] → Model R → R
b2 
2: ?m.2621
2 = 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.2629} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a1: {R : Type ?u.2639} → [inst : IntegralDomain R] → Model R → R
a1 
1: ?m.2643
1 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.2645} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a1: {R : Type ?u.2655} → [inst : IntegralDomain R] → Model R → R
a1 
1: ?m.2659
1 + 
4: ?m.2665
4 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.2674} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a2: {R : Type ?u.2684} → [inst : IntegralDomain R] → Model R → R
a2 
2: ?m.2688
2 := by
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2←
evr: EnatValRing p
evr.
sub_val_mul_right: ∀ {R : Type ?u.3016} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr e.a1 1 * e.a1) 1 + 4 * EnatValRing.sub_val evr e.a2 2 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2←
evr: EnatValRing p
evr.
sub_val_mul_left: ∀ {R : Type ?u.3053} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a1 * e.a1) 1) 1 + 4 * EnatValRing.sub_val evr e.a2 2 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2
evr: EnatValRing p
evr.
sub_val_sub_val: ∀ {R : Type ?u.3071} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_val,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 = EnatValRing.sub_val evr (e.a1 * e.a1) (1 + 1) + 4 * EnatValRing.sub_val evr e.a2 2
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2←
evr: EnatValRing p
evr.
sub_val_mul_right: ∀ {R : Type ?u.3097} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 = EnatValRing.sub_val evr (e.a1 * e.a1) (1 + 1) + EnatValRing.sub_val evr (4 * e.a2) 2 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2←
evr: EnatValRing p
evr.
sub_val_add: ∀ {R : Type ?u.3121} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n →
      EnatValRing.sub_val evr (x + y) n = EnatValRing.sub_val evr x n + EnatValRing.sub_val evr y n
sub_val_add 
_: SurjVal.v evr.valtn ?m.3134 ≥ ofN ?m.3136
_ 
_: SurjVal.v evr.valtn ?m.3135 ≥ ofN ?m.3136
_
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 = EnatValRing.sub_val evr (e.a1 * e.a1 + 4 * e.a2) (1 + 1)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (e.a1 * e.a1) ≥ ofN (1 + 1)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (4 * e.a2) ≥ ofN (1 + 1)]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 = EnatValRing.sub_val evr (e.a1 * e.a1 + 4 * e.a2) (1 + 1)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (e.a1 * e.a1) ≥ ofN (1 + 1)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (4 * e.a2) ≥ ofN (1 + 1)
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 = EnatValRing.sub_val evr (e.a1 * e.a1 + 4 * e.a2) (1 + 1) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 = EnatValRing.sub_val evr (e.a1 * e.a1 + 4 * e.a2) (1 + 1)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (e.a1 * e.a1) ≥ ofN (1 + 1)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (4 * e.a2) ≥ ofN (1 + 1)rfl
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (e.a1 * e.a1) ≥ ofN (1 + 1) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (e.a1 * e.a1) ≥ ofN (1 + 1)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (4 * e.a2) ≥ ofN (1 + 1)exact 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.3170} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.3176} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (4 * e.a2) ≥ ofN (1 + 1) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (4 * e.a2) ≥ ofN (1 + 1)exact 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.3289} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.3294} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2
Goals accomplished! 🐙

lemma 
pi_scaling_of_b4: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
      SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
        EnatValRing.sub_val evr (b4 e) 4 =           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4
pi_scaling_of_b4 (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.3323} → R → [inst : IntegralDomain R] → Type ?u.3323
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.3332) → [inst : IntegralDomain R] → Type ?u.3332
Model 
R: Type u
R) (
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.3337} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.3344} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a1: {R : Type ?u.3351} → [inst : IntegralDomain R] → Model R → R
a1 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.3357
1)
  (
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.3375} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.3379} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a3: {R : Type ?u.3383} → [inst : IntegralDomain R] → Model R → R
a3 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
3: ?m.3387
3) (
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.3397} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.3401} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a4: {R : Type ?u.3405} → [inst : IntegralDomain R] → Model R → R
a4 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
4: ?m.3409
4) :
  
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.3419} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
b4: {R : Type ?u.3429} → [inst : IntegralDomain R] → Model R → R
b4 
4: ?m.3436
4 = 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.3444} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a1: {R : Type ?u.3454} → [inst : IntegralDomain R] → Model R → R
a1 
1: ?m.3458
1 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.3460} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a3: {R : Type ?u.3470} → [inst : IntegralDomain R] → Model R → R
a3 
3: ?m.3474
3 + 
2: ?m.3480
2 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.3489} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a4: {R : Type ?u.3499} → [inst : IntegralDomain R] → Model R → R
a4 
4: ?m.3503
4 := by
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_right: ∀ {R : Type ?u.3832} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr e.a1 1 * e.a3) 3 + 2 * EnatValRing.sub_val evr e.a4 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_left: ∀ {R : Type ?u.3869} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a1 * e.a3) 1) 3 + 2 * EnatValRing.sub_val evr e.a4 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4
evr: EnatValRing p
evr.
sub_val_sub_val: ∀ {R : Type ?u.3887} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_val,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 = EnatValRing.sub_val evr (e.a1 * e.a3) (1 + 3) + 2 * EnatValRing.sub_val evr e.a4 4
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_right: ∀ {R : Type ?u.3913} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 = EnatValRing.sub_val evr (e.a1 * e.a3) (1 + 3) + EnatValRing.sub_val evr (2 * e.a4) 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_add: ∀ {R : Type ?u.3937} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n →
      EnatValRing.sub_val evr (x + y) n = EnatValRing.sub_val evr x n + EnatValRing.sub_val evr y n
sub_val_add 
_: SurjVal.v evr.valtn ?m.3950 ≥ ofN ?m.3952
_ 
_: SurjVal.v evr.valtn ?m.3951 ≥ ofN ?m.3952
_
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 = EnatValRing.sub_val evr (e.a1 * e.a3 + 2 * e.a4) (1 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (e.a1 * e.a3) ≥ ofN (1 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (2 * e.a4) ≥ ofN (1 + 3)]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 = EnatValRing.sub_val evr (e.a1 * e.a3 + 2 * e.a4) (1 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (e.a1 * e.a3) ≥ ofN (1 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (2 * e.a4) ≥ ofN (1 + 3)
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 = EnatValRing.sub_val evr (e.a1 * e.a3 + 2 * e.a4) (1 + 3) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 = EnatValRing.sub_val evr (e.a1 * e.a3 + 2 * e.a4) (1 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (e.a1 * e.a3) ≥ ofN (1 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (2 * e.a4) ≥ ofN (1 + 3)rfl
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (e.a1 * e.a3) ≥ ofN (1 + 3) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (e.a1 * e.a3) ≥ ofN (1 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (2 * e.a4) ≥ ofN (1 + 3)exact 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.3986} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.3992} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (2 * e.a4) ≥ ofN (1 + 3) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (2 * e.a4) ≥ ofN (1 + 3)exact 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.4105} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.4110} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4
Goals accomplished! 🐙

lemma 
pi_scaling_of_b6: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
    SurjVal.v evr.valtn e.a6 ≥ ofN 6 →
      EnatValRing.sub_val evr (b6 e) 6 =         EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6
pi_scaling_of_b6 (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.4140} → R → [inst : IntegralDomain R] → Type ?u.4140
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.4149) → [inst : IntegralDomain R] → Type ?u.4149
Model 
R: Type u
R) (
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.4154} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.4161} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a3: {R : Type ?u.4168} → [inst : IntegralDomain R] → Model R → R
a3 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
3: ?m.4174
3)
  (
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.4191} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.4195} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a6: {R : Type ?u.4199} → [inst : IntegralDomain R] → Model R → R
a6 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
6: ?m.4203
6) :
  
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.4213} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
b6: {R : Type ?u.4223} → [inst : IntegralDomain R] → Model R → R
b6 
6: ?m.4230
6 = 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.4238} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a3: {R : Type ?u.4248} → [inst : IntegralDomain R] → Model R → R
a3 
3: ?m.4252
3 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.4254} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a3: {R : Type ?u.4264} → [inst : IntegralDomain R] → Model R → R
a3 
3: ?m.4268
3 + 
4: ?m.4274
4 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.4283} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a6: {R : Type ?u.4293} → [inst : IntegralDomain R] → Model R → R
a6 
6: ?m.4297
6 := by
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6←
evr: EnatValRing p
evr.
sub_val_mul_right: ∀ {R : Type ?u.4625} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr e.a3 3 * e.a3) 3 + 4 * EnatValRing.sub_val evr e.a6 6 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6←
evr: EnatValRing p
evr.
sub_val_mul_left: ∀ {R : Type ?u.4662} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a3 * e.a3) 3) 3 + 4 * EnatValRing.sub_val evr e.a6 6 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6
evr: EnatValRing p
evr.
sub_val_sub_val: ∀ {R : Type ?u.4680} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_val,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 = EnatValRing.sub_val evr (e.a3 * e.a3) (3 + 3) + 4 * EnatValRing.sub_val evr e.a6 6
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6←
evr: EnatValRing p
evr.
sub_val_mul_right: ∀ {R : Type ?u.4706} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right 
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 = EnatValRing.sub_val evr (e.a3 * e.a3) (3 + 3) + EnatValRing.sub_val evr (4 * e.a6) 6 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6←
evr: EnatValRing p
evr.
sub_val_add: ∀ {R : Type ?u.4730} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n →
      EnatValRing.sub_val evr (x + y) n = EnatValRing.sub_val evr x n + EnatValRing.sub_val evr y n
sub_val_add 
_: SurjVal.v evr.valtn ?m.4743 ≥ ofN ?m.4745
_ 
_: SurjVal.v evr.valtn ?m.4744 ≥ ofN ?m.4745
_
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 = EnatValRing.sub_val evr (e.a3 * e.a3 + 4 * e.a6) (3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (e.a3 * e.a3) ≥ ofN (3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (4 * e.a6) ≥ ofN (3 + 3)]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 = EnatValRing.sub_val evr (e.a3 * e.a3 + 4 * e.a6) (3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (e.a3 * e.a3) ≥ ofN (3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (4 * e.a6) ≥ ofN (3 + 3)
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 = EnatValRing.sub_val evr (e.a3 * e.a3 + 4 * e.a6) (3 + 3) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 = EnatValRing.sub_val evr (e.a3 * e.a3 + 4 * e.a6) (3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (e.a3 * e.a3) ≥ ofN (3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (4 * e.a6) ≥ ofN (3 + 3)rfl
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (e.a3 * e.a3) ≥ ofN (3 + 3) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (e.a3 * e.a3) ≥ ofN (3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (4 * e.a6) ≥ ofN (3 + 3)exact 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.4779} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.4785} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (4 * e.a6) ≥ ofN (3 + 3) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (4 * e.a6) ≥ ofN (3 + 3)exact 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.4898} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.4903} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6
Goals accomplished! 🐙

lemma 
pi_scaling_of_b8: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
          SurjVal.v evr.valtn e.a6 ≥ ofN 6 →
            EnatValRing.sub_val evr (b8 e) 8 =               EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -                       EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +                     4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +                   EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -                 EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
pi_scaling_of_b8 (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.4932} → R → [inst : IntegralDomain R] → Type ?u.4932
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.4941) → [inst : IntegralDomain R] → Type ?u.4941
Model 
R: Type u
R) (
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.4946} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.4953} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a1: {R : Type ?u.4960} → [inst : IntegralDomain R] → Model R → R
a1 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.4966
1)
  (
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.4984} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.4988} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a2: {R : Type ?u.4992} → [inst : IntegralDomain R] → Model R → R
a2 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
2: ?m.4996
2) (
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.5006} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.5010} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a3: {R : Type ?u.5014} → [inst : IntegralDomain R] → Model R → R
a3 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
3: ?m.5018
3) (
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.5028} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.5032} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a4: {R : Type ?u.5036} → [inst : IntegralDomain R] → Model R → R
a4 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
4: ?m.5040
4)
  (
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.5050} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.5054} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a6: {R : Type ?u.5058} → [inst : IntegralDomain R] → Model R → R
a6 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
6: ?m.5062
6) :
  
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.5072} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
b8: {R : Type ?u.5082} → [inst : IntegralDomain R] → Model R → R
b8 
8: ?m.5089
8 = 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.5112} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a1: {R : Type ?u.5122} → [inst : IntegralDomain R] → Model R → R
a1 
1: ?m.5126
1 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.5128} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a1: {R : Type ?u.5138} → [inst : IntegralDomain R] → Model R → R
a1 
1: ?m.5142
1 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.5144} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a6: {R : Type ?u.5154} → [inst : IntegralDomain R] → Model R → R
a6 
6: ?m.5158
6
    - 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.5166} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a1: {R : Type ?u.5176} → [inst : IntegralDomain R] → Model R → R
a1 
1: ?m.5180
1 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.5182} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a3: {R : Type ?u.5192} → [inst : IntegralDomain R] → Model R → R
a3 
3: ?m.5196
3 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.5198} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a4: {R : Type ?u.5208} → [inst : IntegralDomain R] → Model R → R
a4 
4: ?m.5212
4
    + 
4: ?m.5221
4 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.5230} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a2: {R : Type ?u.5240} → [inst : IntegralDomain R] → Model R → R
a2 
2: ?m.5244
2 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.5246} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a6: {R : Type ?u.5256} → [inst : IntegralDomain R] → Model R → R
a6 
6: ?m.5260
6
    + 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.5268} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a2: {R : Type ?u.5278} → [inst : IntegralDomain R] → Model R → R
a2 
2: ?m.5282
2 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.5284} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a3: {R : Type ?u.5294} → [inst : IntegralDomain R] → Model R → R
a3 
3: ?m.5298
3 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.5300} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a3: {R : Type ?u.5310} → [inst : IntegralDomain R] → Model R → R
a3 
3: ?m.5314
3
    - 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.5319} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a4: {R : Type ?u.5329} → [inst : IntegralDomain R] → Model R → R
a4 
4: ?m.5333
4 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.5335} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
a4: {R : Type ?u.5345} → [inst : IntegralDomain R] → Model R → R
a4 
4: ?m.5349
4 :=
by
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_right: ∀ {R : Type ?u.6209} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr e.a1 1 * e.a1) 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_left: ∀ {R : Type ?u.6264} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a1 * e.a1) 1) 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
evr: EnatValRing p
evr.
sub_val_sub_val: ∀ {R : Type ?u.6282} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_val,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1) (1 + 1) * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_right: ∀ {R : Type ?u.6302} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right 
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a1 * e.a1) (1 + 1) * e.a6) 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_left: ∀ {R : Type ?u.6320} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.6335} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.6341} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1),
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1)) 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
evr: EnatValRing p
evr.
sub_val_sub_val: ∀ {R : Type ?u.6451} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_val
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_right: ∀ {R : Type ?u.6478} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr (EnatValRing.sub_val evr e.a1 1 * e.a3) 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_left: ∀ {R : Type ?u.6496} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a1 * e.a3) 1) 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
evr: EnatValRing p
evr.
sub_val_sub_val: ∀ {R : Type ?u.6514} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_val,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr (e.a1 * e.a3) (1 + 3) * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_right: ∀ {R : Type ?u.6531} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a1 * e.a3) (1 + 3) * e.a4) 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_left: ∀ {R : Type ?u.6549} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.6564} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.6570} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3),
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3)) 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
evr: EnatValRing p
evr.
sub_val_sub_val: ∀ {R : Type ?u.6666} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_val
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_right: ∀ {R : Type ?u.6687} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2) 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_right: ∀ {R : Type ?u.6705} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right 
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (EnatValRing.sub_val evr (4 * e.a2) 2 * e.a6) 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_left: ∀ {R : Type ?u.6723} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left (
val_mul_ge_of_right_ge: ∀ {R : Type ?u.6738} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.6743} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2),
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (EnatValRing.sub_val evr (4 * e.a2 * e.a6) 2) 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
evr: EnatValRing p
evr.
sub_val_sub_val: ∀ {R : Type ?u.6755} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_val
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_left: ∀ {R : Type ?u.6776} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * EnatValRing.sub_val evr e.a3 3) 2 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_right: ∀ {R : Type ?u.6794} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a2 * EnatValRing.sub_val evr e.a3 3) 2 * e.a3) 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_right: ∀ {R : Type ?u.6812} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a2 * e.a3) 3) 2 * e.a3) 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
evr: EnatValRing p
evr.
sub_val_sub_val: ∀ {R : Type ?u.6830} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_val,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a2 * e.a3) (3 + 2) * e.a3) 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_left: ∀ {R : Type ?u.6847} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.6862} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.6868} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3),
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3)) 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
evr: EnatValRing p
evr.
sub_val_sub_val: ∀ {R : Type ?u.6980} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_val
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_right: ∀ {R : Type ?u.7001} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (EnatValRing.sub_val evr e.a4 4 * e.a4) 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←
evr: EnatValRing p
evr.
sub_val_mul_left: ∀ {R : Type ?u.7019} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a4 * e.a4) 4) 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
evr: EnatValRing p
evr.
sub_val_sub_val: ∀ {R : Type ?u.7037} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_val
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4have 
h116: ?m.7059
h116 := 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.7060} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.7066} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.7075} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.7081} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1) 
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN 1 + ofN 1 + ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4have 
h134: ?m.7095
h134 := (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.7096} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.7102} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.7111} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.7117} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3) 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN 1 + ofN 1 + ofN 6

h134: SurjVal.v evr.valtn (e.a1 * e.a3 * e.a4) ≥ ofN 1 + ofN 3 + ofN 4

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4have 
h26: ?m.7131
h26 := 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.7132} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.7138} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (@
val_mul_ge_of_right_ge: ∀ {R : Type ?u.7147} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
R: Type u
R _ (
ofN: ℕ → ℕ∪∞
ofN 
2: ?m.7150
2) 
p: R
p 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.7152} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
4: ?m.7157
4 
e: Model R
e.
a2: {R : Type ?u.7159} → [inst : IntegralDomain R] → Model R → R
a2 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2) 
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN 1 + ofN 1 + ofN 6

h134: SurjVal.v evr.valtn (e.a1 * e.a3 * e.a4) ≥ ofN 1 + ofN 3 + ofN 4

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 2 + ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4have 
h233: ?m.7167
h233 := 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.7168} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.7174} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.7183} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.7189} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3) 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN 1 + ofN 1 + ofN 6

h134: SurjVal.v evr.valtn (e.a1 * e.a3 * e.a4) ≥ ofN 1 + ofN 3 + ofN 4

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 2 + ofN 6

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN 2 + ofN 3 + ofN 3

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4have 
h44: ?m.7203
h44 := 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.7204} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.7210} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN 1 + ofN 1 + ofN 6

h134: SurjVal.v evr.valtn (e.a1 * e.a3 * e.a4) ≥ ofN 1 + ofN 3 + ofN 4

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 2 + ofN 6

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN 2 + ofN 3 + ofN 3

h44: SurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN 4 + ofN 4

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4simp only [
add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN] at 
h116: ?m.7059
h116 
h134: ?m.7095
h134 
h44: ?m.7203
h44 
h26: ?m.7131
h26 
h233: ?m.7167
h233
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (e.a1 * e.a3 * e.a4) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (e.a1 * e.a3 * e.a4) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←
val_neg: ∀ {R : Type ?u.7397} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4) at 
h134: SurjVal.v evr.valtn (e.a1 * e.a3 * e.a4) ≥ ofN (1 + 3 + 4)
h134
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←
val_neg: ∀ {R : Type ?u.7431} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4) at 
h44: SurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN (4 + 4)
h44
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)

  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)
sub_eq_add_neg: ∀ {G : Type ?u.7460} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_neg,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) +     -EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)
sub_eq_add_neg: ∀ {G : Type ?u.7582} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_neg,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) + -EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) +     -EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←
evr: EnatValRing p
evr.
sub_val_neg: ∀ {R : Type ?u.7652} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {n : ℕ},
  EnatValRing.sub_val evr (-x) n = -EnatValRing.sub_val evr x n
sub_val_neg,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) +           EnatValRing.sub_val evr (-(e.a1 * e.a3 * e.a4)) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) +     -EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←
evr: EnatValRing p
evr.
sub_val_neg: ∀ {R : Type ?u.7670} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {n : ℕ},
  EnatValRing.sub_val evr (-x) n = -EnatValRing.sub_val evr x n
sub_val_neg,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) +           EnatValRing.sub_val evr (-(e.a1 * e.a3 * e.a4)) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) +     EnatValRing.sub_val evr (-(e.a4 * e.a4)) (4 + 4)
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←
evr: EnatValRing p
evr.
sub_val_add: ∀ {R : Type ?u.7688} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n →
      EnatValRing.sub_val evr (x + y) n = EnatValRing.sub_val evr x n + EnatValRing.sub_val evr y n
sub_val_add 
h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)
h116 
h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)
h134,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) (1 + 1 + 6) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) +     EnatValRing.sub_val evr (-(e.a4 * e.a4)) (4 + 4) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←
evr: EnatValRing p
evr.
sub_val_add: ∀ {R : Type ?u.7724} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n →
      EnatValRing.sub_val evr (x + y) n = EnatValRing.sub_val evr x n + EnatValRing.sub_val evr y n
sub_val_add 
_: SurjVal.v evr.valtn ?m.7737 ≥ ofN ?m.7739
_ 
h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)
h26,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) +     EnatValRing.sub_val evr (-(e.a4 * e.a4)) (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←
evr: EnatValRing p
evr.
sub_val_add: ∀ {R : Type ?u.7750} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n →
      EnatValRing.sub_val evr (x + y) n = EnatValRing.sub_val evr x n + EnatValRing.sub_val evr y n
sub_val_add 
_: SurjVal.v evr.valtn ?m.7763 ≥ ofN ?m.7765
_ 
h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)
h233,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3)
      (2 + 3 + 3) +     EnatValRing.sub_val evr (-(e.a4 * e.a4)) (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN (2 + 3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6)
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←
evr: EnatValRing p
evr.
sub_val_add: ∀ {R : Type ?u.7770} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n →
      EnatValRing.sub_val evr (x + y) n = EnatValRing.sub_val evr x n + EnatValRing.sub_val evr y n
sub_val_add 
_: SurjVal.v evr.valtn ?m.7783 ≥ ofN ?m.7785
_ 
h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)
h44,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr
    (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 + -(e.a4 * e.a4)) (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN (2 + 3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←
sub_eq_add_neg: ∀ {G : Type ?u.7790} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_neg,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr
    (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 - e.a4 * e.a4) (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN (2 + 3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←
sub_eq_add_neg: ∀ {G : Type ?u.7858} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_neg
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6 - e.a1 * e.a3 * e.a4 + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 - e.a4 * e.a4)
    (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN (2 + 3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6)]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6 - e.a1 * e.a3 * e.a4 + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 - e.a4 * e.a4)
    (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN (2 + 3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6)
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6 - e.a1 * e.a3 * e.a4 + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 - e.a4 * e.a4)
    (4 + 4) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6 - e.a1 * e.a3 * e.a4 + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 - e.a4 * e.a4)
    (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN (2 + 3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6)rfl
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN (4 + 4) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN (2 + 3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6)exact 
val_add_ge_of_ge: ∀ {R : Type ?u.7958} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.7963} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn
      (
val_add_ge_of_ge: ∀ {R : Type ?u.7978} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.7983} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (
val_add_ge_of_ge: ∀ {R : Type ?u.7992} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.7997} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)
h116 
h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)
h134) 
h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)
h26) 
h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)
h233
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN (2 + 3 + 3) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN (2 + 3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6)exact 
val_add_ge_of_ge: ∀ {R : Type ?u.8006} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.8011} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (
val_add_ge_of_ge: ∀ {R : Type ?u.8020} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.8025} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)
h116 
h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)
h134) 
h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)
h26
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6)exact 
val_add_ge_of_ge: ∀ {R : Type ?u.8034} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.8039} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)
h116 
h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)
h134
Goals accomplished! 🐙

open EnatValRing in
lemma 
pi_scaling_of_discr: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn (b2 e) ≥ ofN 2 →
    SurjVal.v evr.valtn (b4 e) ≥ ofN 4 →
      SurjVal.v evr.valtn (b6 e) ≥ ofN 6 →
        SurjVal.v evr.valtn (b8 e) ≥ ofN 8 →
          sub_val evr (discr e) 12 =             -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -                 27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +               9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
pi_scaling_of_discr (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.8108} → R → [inst : IntegralDomain R] → Type ?u.8108
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.8117) → [inst : IntegralDomain R] → Type ?u.8117
Model 
R: Type u
R)
  (
hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.8122} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.8129} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
b2: {R : Type ?u.8136} → [inst : IntegralDomain R] → Model R → R
b2 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
2: ?m.8145
2) (
hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4
hb4 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.8162} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.8166} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
b4: {R : Type ?u.8170} → [inst : IntegralDomain R] → Model R → R
b4 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
4: ?m.8177
4)
  (
hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.8187} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.8191} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
b6: {R : Type ?u.8195} → [inst : IntegralDomain R] → Model R → R
b6 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
6: ?m.8202
6) (
hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8
hb8 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.8212} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.8216} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
b8: {R : Type ?u.8220} → [inst : IntegralDomain R] → Model R → R
b8 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
8: ?m.8227
8) :
  
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.8237} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
discr: {R : Type ?u.8247} → [inst : IntegralDomain R] → Model R → R
discr 
12: ?m.8254
12 = -
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.8275} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
b2: {R : Type ?u.8285} → [inst : IntegralDomain R] → Model R → R
b2 
2: ?m.8292
2 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.8294} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
b2: {R : Type ?u.8304} → [inst : IntegralDomain R] → Model R → R
b2 
2: ?m.8311
2 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.8313} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
b8: {R : Type ?u.8323} → [inst : IntegralDomain R] → Model R → R
b8 
8: ?m.8330
8
    - 
8: ?m.8336
8 * ((
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.8348} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
b4: {R : Type ?u.8358} → [inst : IntegralDomain R] → Model R → R
b4 
4: ?m.8365
4) ^ 
3: ?m.8368
3) - 
27: ?m.8384
27 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.8393} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
b6: {R : Type ?u.8403} → [inst : IntegralDomain R] → Model R → R
b6 
6: ?m.8410
6 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.8412} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
b6: {R : Type ?u.8422} → [inst : IntegralDomain R] → Model R → R
b6 
6: ?m.8429
6
    + 
9: ?m.8441
9 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.8450} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
b2: {R : Type ?u.8460} → [inst : IntegralDomain R] → Model R → R
b2 
2: ?m.8467
2 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.8469} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
b4: {R : Type ?u.8479} → [inst : IntegralDomain R] → Model R → R
b4 
4: ?m.8486
4 * 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.8488} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
b6: {R : Type ?u.8498} → [inst : IntegralDomain R] → Model R → R
b6 
6: ?m.8505
6 :=
by
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
discr: {R : Type ?u.9642} → [inst : IntegralDomain R] → Model R → R
discr,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e + 9 * b2 e * b4 e * b6 e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
sub_val_add: ∀ {R : Type ?u.9643} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n → sub_val evr (x + y) n = sub_val evr x n + sub_val evr y n
sub_val_add,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) 12 + sub_val evr (9 * b2 e * b4 e * b6 e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
sub_val_sub: ∀ {R : Type ?u.9694} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n → sub_val evr (x - y) n = sub_val evr x n - sub_val evr y n
sub_val_sub,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) 12 - sub_val evr (27 * b6 e * b6 e) 12 +     sub_val evr (9 * b2 e * b4 e * b6 e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
sub_val_sub: ∀ {R : Type ?u.9730} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n → sub_val evr (x - y) n = sub_val evr x n - sub_val evr y n
sub_val_sub,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e * b2 e * b8 e) 12 - sub_val evr (8 * b4 e ^ 3) 12 - sub_val evr (27 * b6 e * b6 e) 12 +     sub_val evr (9 * b2 e * b4 e * b6 e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
sub_val_mul: ∀ {R : Type ?u.9754} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} (n m : ℕ) {nm : ℕ},
  n + m = nm →
    SurjVal.v evr.valtn x ≥ ofN n →
      SurjVal.v evr.valtn y ≥ ofN m → sub_val evr (x * y) nm = sub_val evr x n * sub_val evr y m
sub_val_mul 
_: EnatValRing ?m.9757
_ 
4: ?m.9762
4 
8: ?m.9767
8 
_: 4 + 8 = ?m.9769
_ 
_: SurjVal.v ?m.9758.valtn ?m.9759 ≥ ofN 4
_ 
hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8
hb8,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e * b2 e) 4 * sub_val evr (b8 e) 8 - sub_val evr (8 * b4 e ^ 3) 12 -       sub_val evr (27 * b6 e * b6 e) 12 +     sub_val evr (9 * b2 e * b4 e * b6 e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
sub_val_mul: ∀ {R : Type ?u.9797} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} (n m : ℕ) {nm : ℕ},
  n + m = nm →
    SurjVal.v evr.valtn x ≥ ofN n →
      SurjVal.v evr.valtn y ≥ ofN m → sub_val evr (x * y) nm = sub_val evr x n * sub_val evr y m
sub_val_mul 
_: EnatValRing ?m.9800
_ 
2: ?m.9805
2 
2: ?m.9808
2 
_: 2 + 2 = ?m.9810
_ 
_: SurjVal.v ?m.9801.valtn ?m.9802 ≥ ofN 2
_ 
hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - sub_val evr (8 * b4 e ^ 3) 12 -       sub_val evr (27 * b6 e * b6 e) 12 +     sub_val evr (9 * b2 e * b4 e * b6 e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
sub_val_mul: ∀ {R : Type ?u.9821} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} (n m : ℕ) {nm : ℕ},
  n + m = nm →
    SurjVal.v evr.valtn x ≥ ofN n →
      SurjVal.v evr.valtn y ≥ ofN m → sub_val evr (x * y) nm = sub_val evr x n * sub_val evr y m
sub_val_mul 
_: EnatValRing ?m.9824
_ 
6: ?m.9829
6 
6: ?m.9832
6 
_: 6 + 6 = ?m.9834
_ 
_: SurjVal.v ?m.9825.valtn ?m.9826 ≥ ofN 6
_ 
hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - sub_val evr (8 * b4 e ^ 3) 12 -       sub_val evr (27 * b6 e) 6 * sub_val evr (b6 e) 6 +     sub_val evr (9 * b2 e * b4 e * b6 e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
sub_val_mul_right: ∀ {R : Type ?u.9860} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → sub_val evr (x * y) n = x * sub_val evr y n
sub_val_mul_right 
_: EnatValRing ?m.9863
_ 
hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - sub_val evr (8 * b4 e ^ 3) 12 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     sub_val evr (9 * b2 e * b4 e * b6 e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
sub_val_mul: ∀ {R : Type ?u.9877} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} (n m : ℕ) {nm : ℕ},
  n + m = nm →
    SurjVal.v evr.valtn x ≥ ofN n →
      SurjVal.v evr.valtn y ≥ ofN m → sub_val evr (x * y) nm = sub_val evr x n * sub_val evr y m
sub_val_mul 
_: EnatValRing ?m.9880
_ 
6: ?m.9885
6 
6: ?m.9888
6 
_: 6 + 6 = ?m.9890
_ 
_: SurjVal.v ?m.9881.valtn ?m.9882 ≥ ofN 6
_ 
hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - sub_val evr (8 * b4 e ^ 3) 12 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     sub_val evr (9 * b2 e * b4 e) 6 * sub_val evr (b6 e) 6 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
sub_val_mul: ∀ {R : Type ?u.9901} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} (n m : ℕ) {nm : ℕ},
  n + m = nm →
    SurjVal.v evr.valtn x ≥ ofN n →
      SurjVal.v evr.valtn y ≥ ofN m → sub_val evr (x * y) nm = sub_val evr x n * sub_val evr y m
sub_val_mul 
_: EnatValRing ?m.9904
_ 
2: ?m.9909
2 
4: ?m.9912
4 
_: 2 + 4 = ?m.9914
_ 
_: SurjVal.v ?m.9905.valtn ?m.9906 ≥ ofN 2
_ 
hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4
hb4,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - sub_val evr (8 * b4 e ^ 3) 12 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     sub_val evr (9 * b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
sub_val_mul_right: ∀ {R : Type ?u.9925} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → sub_val evr (x * y) n = x * sub_val evr y n
sub_val_mul_right 
_: EnatValRing ?m.9928
_ 
hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - sub_val evr (8 * b4 e ^ 3) 12 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
sub_val_mul_right: ∀ {R : Type ?u.9941} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → sub_val evr (x * y) n = x * sub_val evr y n
sub_val_mul_right,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e ^ 3) 12 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
sub_val_pow: ∀ {R : Type ?u.9963} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} (n k : ℕ) {nm : ℕ},
  k * n = nm → SurjVal.v evr.valtn x ≥ ofN n → sub_val evr (x ^ k) nm = sub_val evr x n ^ k
sub_val_pow 
_: EnatValRing ?m.9966
_ 
_: ℕ
_ 
_: ℕ
_ 
_: ?m.9970 * ?m.9969 = ?m.9971
_ 
hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4
hb4,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

3 * 4 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
sub_val_neg: ∀ {R : Type ?u.9993} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {n : ℕ},
  sub_val evr (-x) n = -sub_val evr x n
sub_val_neg
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

-sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

3 * 4 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

3 * 4 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

3 * 4 = 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

3 * 4 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12rfl
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (b4 e ^ 3) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12exact 
val_pow_ge_of_ge: ∀ {R : Type ?u.10029} [inst : IntegralDomain R] {m : ℕ∪∞} {p : R} (nav : SurjVal p) {a : R} (k : ℕ),
  SurjVal.v nav a ≥ m → SurjVal.v nav (a ^ k) ≥ k • m
val_pow_ge_of_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.10034} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
3: ?m.10046
3 
hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4
hb4
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12norm_num
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12exact 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.10180} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
_: SurjVal ?m.10184
_ 
hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12norm_num
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12exact 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.10204} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.10210} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (
val_mul_ge_of_right_ge: ∀ {R : Type ?u.10258} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.10263} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2) 
hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4
hb4
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12norm_num
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12exact 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.10319} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.10324} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12norm_num
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12simpa
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12norm_num
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12exact 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.10623} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.10629} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4by 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2simpa
Goals accomplished! 🐙 : SurjVal.v evr.valtn (-b2 e) ≥ ofN 2) 
hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12exact 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.10945} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.10951} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn
      (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.10999} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.11005} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12by 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2simpa
Goals accomplished! 🐙 : SurjVal.v evr.valtn (-b2 e) ≥ ofN 2) 
hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2) 
hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8
hb8
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12exact 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.11304} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.11309} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn $ 
val_pow_ge_of_ge: ∀ {R : Type ?u.11318} [inst : IntegralDomain R] {m : ℕ∪∞} {p : R} (nav : SurjVal p) {a : R} (k : ℕ),
  SurjVal.v nav a ≥ m → SurjVal.v nav (a ^ k) ≥ k • m
val_pow_ge_of_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.11323} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
3: ?m.11332
3 
hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4
hb4
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12apply 
val_sub_ge_of_ge: ∀ {R : Type ?u.11413} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a - b) ≥ n
val_sub_ge_of_ge
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hb
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hb
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12exact 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.11443} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.11449} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn
        (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.11497} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.11503} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12by 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2simpa
Goals accomplished! 🐙 : SurjVal.v evr.valtn (-b2 e) ≥ ofN 2) 
hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2) 
hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8
hb8
Goals accomplished! 🐙
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hb
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hb
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12exact 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.11802} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.11807} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (
val_pow_ge_of_ge: ∀ {R : Type ?u.11816} [inst : IntegralDomain R] {m : ℕ∪∞} {p : R} (nav : SurjVal p) {a : R} (k : ℕ),
  SurjVal.v nav a ≥ m → SurjVal.v nav (a ^ k) ≥ k • m
val_pow_ge_of_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.11821} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
3: ?m.11830
3 
hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4
hb4)
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12exact 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.11911} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.11917} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (
val_mul_ge_of_right_ge: ∀ {R : Type ?u.11965} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.11970} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6) 
hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12apply 
val_sub_ge_of_ge: ∀ {R : Type ?u.12024} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a - b) ≥ n
val_sub_ge_of_ge
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hb
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hb
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12apply 
val_sub_ge_of_ge: ∀ {R : Type ?u.12048} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a - b) ≥ n
val_sub_ge_of_ge
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha.hb
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hb
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12exact 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.12072} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.12078} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn
        (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.12126} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.12132} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha.hb
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12by 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2simpa
Goals accomplished! 🐙 : SurjVal.v evr.valtn (-b2 e) ≥ ofN 2) 
hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2) 
hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8
hb8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha.hb
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hb
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12apply 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.12431} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.12436} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha.hb
SurjVal.v evr.valtn (b4 e ^ 3) ≥ ofN 12
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hb
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12exact 
val_pow_ge_of_ge: ∀ {R : Type ?u.12448} [inst : IntegralDomain R] {m : ℕ∪∞} {p : R} (nav : SurjVal p) {a : R} (k : ℕ),
  SurjVal.v nav a ≥ m → SurjVal.v nav (a ^ k) ≥ k • m
val_pow_ge_of_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.12453} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
3: ?m.12462
3 
hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4
hb4
Goals accomplished! 🐙
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hb
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hb
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12exact 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.12543} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.12549} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (
val_mul_ge_of_right_ge: ∀ {R : Type ?u.12597} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.12602} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6) 
hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hy
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12exact 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.12656} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge
      
evr: EnatValRing p
evr.
valtn: {R : Type ?u.12662} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.12710} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.12716} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (
val_mul_ge_of_right_ge: ∀ {R : Type ?u.12725} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.12730} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2) 
hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4
hb4) 
hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6
Goals accomplished! 🐙

lemma 
b2_of_pi_scaling: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 → b2 (pi_scaling evr e) = EnatValRing.sub_val evr (b2 e) 2
b2_of_pi_scaling (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.12841} → R → [inst : IntegralDomain R] → Type ?u.12841
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.12850) → [inst : IntegralDomain R] → Type ?u.12850
Model 
R: Type u
R) (
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.12855} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.12862} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a1: {R : Type ?u.12869} → [inst : IntegralDomain R] → Model R → R
a1 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.12875
1)
  (
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.12893} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.12897} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a2: {R : Type ?u.12901} → [inst : IntegralDomain R] → Model R → R
a2 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
2: ?m.12905
2) : (
pi_scaling: {R : Type ?u.12915} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling 
evr: EnatValRing p
evr 
e: Model R
e).
b2: {R : Type ?u.12922} → [inst : IntegralDomain R] → Model R → R
b2 = 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.12928} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
b2: {R : Type ?u.12938} → [inst : IntegralDomain R] → Model R → R
b2 
2: ?m.12945
2 := by
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

b2 (pi_scaling evr e) = EnatValRing.sub_val evr (b2 e) 2simp only [
b2: {R : Type ?u.12962} → [inst : IntegralDomain R] → Model R → R
b2, 
pi_scaling: {R : Type ?u.12967} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2 =   EnatValRing.sub_val evr (e.a1 * e.a1 + 4 * e.a2) 2
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

b2 (pi_scaling evr e) = EnatValRing.sub_val evr (b2 e) 2exact (
pi_scaling_of_b2: ∀ {R : Type ?u.13159} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      EnatValRing.sub_val evr (b2 e) 2 =         EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2
pi_scaling_of_b2 
evr: EnatValRing p
evr 
e: Model R
e 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2).
symm: ∀ {α : Sort ?u.13170} {a b : α}, a = b → b = a
symm
Goals accomplished! 🐙

lemma 
b4_of_pi_scaling: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
      SurjVal.v evr.valtn e.a4 ≥ ofN 4 → b4 (pi_scaling evr e) = EnatValRing.sub_val evr (b4 e) 4
b4_of_pi_scaling (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.13210} → R → [inst : IntegralDomain R] → Type ?u.13210
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.13219) → [inst : IntegralDomain R] → Type ?u.13219
Model 
R: Type u
R) (
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.13224} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.13231} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a1: {R : Type ?u.13238} → [inst : IntegralDomain R] → Model R → R
a1 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.13244
1)
  (
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.13262} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.13266} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a3: {R : Type ?u.13270} → [inst : IntegralDomain R] → Model R → R
a3 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
3: ?m.13274
3) (
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.13284} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.13288} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a4: {R : Type ?u.13292} → [inst : IntegralDomain R] → Model R → R
a4 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
4: ?m.13296
4) :
  (
pi_scaling: {R : Type ?u.13306} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling 
evr: EnatValRing p
evr 
e: Model R
e).
b4: {R : Type ?u.13313} → [inst : IntegralDomain R] → Model R → R
b4 = 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.13319} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
b4: {R : Type ?u.13329} → [inst : IntegralDomain R] → Model R → R
b4 
4: ?m.13336
4 := by
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

b4 (pi_scaling evr e) = EnatValRing.sub_val evr (b4 e) 4simp only [
b4: {R : Type ?u.13354} → [inst : IntegralDomain R] → Model R → R
b4, 
pi_scaling: {R : Type ?u.13359} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4 =   EnatValRing.sub_val evr (e.a1 * e.a3 + 2 * e.a4) 4
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

b4 (pi_scaling evr e) = EnatValRing.sub_val evr (b4 e) 4exact (
pi_scaling_of_b4: ∀ {R : Type ?u.13551} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
      SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
        EnatValRing.sub_val evr (b4 e) 4 =           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4
pi_scaling_of_b4 
evr: EnatValRing p
evr 
e: Model R
e 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4).
symm: ∀ {α : Sort ?u.13562} {a b : α}, a = b → b = a
symm
Goals accomplished! 🐙

lemma 
b6_of_pi_scaling: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
    SurjVal.v evr.valtn e.a6 ≥ ofN 6 → b6 (pi_scaling evr e) = EnatValRing.sub_val evr (b6 e) 6
b6_of_pi_scaling (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.13603} → R → [inst : IntegralDomain R] → Type ?u.13603
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.13612) → [inst : IntegralDomain R] → Type ?u.13612
Model 
R: Type u
R) (
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.13617} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.13624} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a3: {R : Type ?u.13631} → [inst : IntegralDomain R] → Model R → R
a3 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
3: ?m.13637
3)
  (
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.13654} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.13658} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a6: {R : Type ?u.13662} → [inst : IntegralDomain R] → Model R → R
a6 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
6: ?m.13666
6) :
  (
pi_scaling: {R : Type ?u.13676} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling 
evr: EnatValRing p
evr 
e: Model R
e).
b6: {R : Type ?u.13683} → [inst : IntegralDomain R] → Model R → R
b6 = 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.13689} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
b6: {R : Type ?u.13699} → [inst : IntegralDomain R] → Model R → R
b6 
6: ?m.13706
6 := by
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

b6 (pi_scaling evr e) = EnatValRing.sub_val evr (b6 e) 6simp only [
b6: {R : Type ?u.13723} → [inst : IntegralDomain R] → Model R → R
b6, 
pi_scaling: {R : Type ?u.13728} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6 =   EnatValRing.sub_val evr (e.a3 * e.a3 + 4 * e.a6) 6
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

b6 (pi_scaling evr e) = EnatValRing.sub_val evr (b6 e) 6exact (
pi_scaling_of_b6: ∀ {R : Type ?u.13920} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
    SurjVal.v evr.valtn e.a6 ≥ ofN 6 →
      EnatValRing.sub_val evr (b6 e) 6 =         EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6
pi_scaling_of_b6 
evr: EnatValRing p
evr 
e: Model R
e 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6).
symm: ∀ {α : Sort ?u.13931} {a b : α}, a = b → b = a
symm
Goals accomplished! 🐙

lemma 
b8_of_pi_scaling: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
          SurjVal.v evr.valtn e.a6 ≥ ofN 6 → b8 (pi_scaling evr e) = EnatValRing.sub_val evr (b8 e) 8
b8_of_pi_scaling (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.13971} → R → [inst : IntegralDomain R] → Type ?u.13971
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.13980) → [inst : IntegralDomain R] → Type ?u.13980
Model 
R: Type u
R) (
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.13985} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.13992} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a1: {R : Type ?u.13999} → [inst : IntegralDomain R] → Model R → R
a1 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.14005
1)
  (
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.14023} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.14027} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a2: {R : Type ?u.14031} → [inst : IntegralDomain R] → Model R → R
a2 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
2: ?m.14035
2) (
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.14045} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.14049} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a3: {R : Type ?u.14053} → [inst : IntegralDomain R] → Model R → R
a3 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
3: ?m.14057
3) (
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.14067} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.14071} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a4: {R : Type ?u.14075} → [inst : IntegralDomain R] → Model R → R
a4 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
4: ?m.14079
4)
  (
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.14089} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.14093} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a6: {R : Type ?u.14097} → [inst : IntegralDomain R] → Model R → R
a6 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
6: ?m.14101
6) : (
pi_scaling: {R : Type ?u.14111} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling 
evr: EnatValRing p
evr 
e: Model R
e).
b8: {R : Type ?u.14118} → [inst : IntegralDomain R] → Model R → R
b8 = 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.14124} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
b8: {R : Type ?u.14134} → [inst : IntegralDomain R] → Model R → R
b8 
8: ?m.14141
8 := by
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

b8 (pi_scaling evr e) = EnatValRing.sub_val evr (b8 e) 8simp only [
b8: {R : Type ?u.14164} → [inst : IntegralDomain R] → Model R → R
b8, 
pi_scaling: {R : Type ?u.14169} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6 - e.a1 * e.a3 * e.a4 + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 - e.a4 * e.a4)
    8
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

b8 (pi_scaling evr e) = EnatValRing.sub_val evr (b8 e) 8exact (
pi_scaling_of_b8: ∀ {R : Type ?u.14391} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
          SurjVal.v evr.valtn e.a6 ≥ ofN 6 →
            EnatValRing.sub_val evr (b8 e) 8 =               EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -                       EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +                     4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +                   EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -                 EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
pi_scaling_of_b8 
evr: EnatValRing p
evr 
e: Model R
e 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6).
symm: ∀ {α : Sort ?u.14402} {a b : α}, a = b → b = a
symm
Goals accomplished! 🐙

lemma 
val_b2_of_val_a12: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 → SurjVal.v evr.valtn e.a2 ≥ ofN 2 → SurjVal.v evr.valtn (b2 e) ≥ ofN 2
val_b2_of_val_a12 (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.14447} → R → [inst : IntegralDomain R] → Type ?u.14447
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.14456) → [inst : IntegralDomain R] → Type ?u.14456
Model 
R: Type u
R) (
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.14461} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.14468} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a1: {R : Type ?u.14475} → [inst : IntegralDomain R] → Model R → R
a1 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.14481
1)
  (
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.14499} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.14503} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a2: {R : Type ?u.14507} → [inst : IntegralDomain R] → Model R → R
a2 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
2: ?m.14511
2) : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.14521} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.14525} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
b2: {R : Type ?u.14529} → [inst : IntegralDomain R] → Model R → R
b2 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
2: ?m.14536
2 := by
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (b2 e) ≥ ofN 2simp only [
b2: {R : Type ?u.14558} → [inst : IntegralDomain R] → Model R → R
b2]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (e.a1 * e.a1 + 4 * e.a2) ≥ ofN 2
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (b2 e) ≥ ofN 2apply 
val_add_ge_of_ge: ∀ {R : Type ?u.14717} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

ha
SurjVal.v evr.valtn (e.a1 * e.a1) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

hb
SurjVal.v evr.valtn (4 * e.a2) ≥ ofN 2
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (b2 e) ≥ ofN 2.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

ha
SurjVal.v evr.valtn (e.a1 * e.a1) ≥ ofN 2 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

ha
SurjVal.v evr.valtn (e.a1 * e.a1) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

hb
SurjVal.v evr.valtn (4 * e.a2) ≥ ofN 2apply 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.14751} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.14757} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (b2 e) ≥ ofN 2.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

hb
SurjVal.v evr.valtn (4 * e.a2) ≥ ofN 2 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

hb
SurjVal.v evr.valtn (4 * e.a2) ≥ ofN 2apply 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.14781} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.14786} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2
Goals accomplished! 🐙

lemma 
val_b4_of_val_a134: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a3 ≥ ofN 3 → SurjVal.v evr.valtn e.a4 ≥ ofN 4 → SurjVal.v evr.valtn (b4 e) ≥ ofN 4
val_b4_of_val_a134 (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.14816} → R → [inst : IntegralDomain R] → Type ?u.14816
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.14825) → [inst : IntegralDomain R] → Type ?u.14825
Model 
R: Type u
R) (
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.14830} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.14837} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a1: {R : Type ?u.14844} → [inst : IntegralDomain R] → Model R → R
a1 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.14850
1)
  (
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.14868} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.14872} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a3: {R : Type ?u.14876} → [inst : IntegralDomain R] → Model R → R
a3 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
3: ?m.14880
3) (
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.14890} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.14894} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a4: {R : Type ?u.14898} → [inst : IntegralDomain R] → Model R → R
a4 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
4: ?m.14902
4) : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.14912} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.14916} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
b4: {R : Type ?u.14920} → [inst : IntegralDomain R] → Model R → R
b4 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
4: ?m.14927
4 := by
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (b4 e) ≥ ofN 4simp only [
b4: {R : Type ?u.14951} → [inst : IntegralDomain R] → Model R → R
b4]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (e.a1 * e.a3 + 2 * e.a4) ≥ ofN 4
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (b4 e) ≥ ofN 4apply 
val_add_ge_of_ge: ∀ {R : Type ?u.15116} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

ha
SurjVal.v evr.valtn (e.a1 * e.a3) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

hb
SurjVal.v evr.valtn (2 * e.a4) ≥ ofN 4
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (b4 e) ≥ ofN 4.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

ha
SurjVal.v evr.valtn (e.a1 * e.a3) ≥ ofN 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

ha
SurjVal.v evr.valtn (e.a1 * e.a3) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

hb
SurjVal.v evr.valtn (2 * e.a4) ≥ ofN 4apply 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.15150} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.15156} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (b4 e) ≥ ofN 4.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

hb
SurjVal.v evr.valtn (2 * e.a4) ≥ ofN 4 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

hb
SurjVal.v evr.valtn (2 * e.a4) ≥ ofN 4apply 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.15180} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.15185} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4
Goals accomplished! 🐙

lemma 
val_b6_of_val_a36: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a3 ≥ ofN 3 → SurjVal.v evr.valtn e.a6 ≥ ofN 6 → SurjVal.v evr.valtn (b6 e) ≥ ofN 6
val_b6_of_val_a36 (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.15216} → R → [inst : IntegralDomain R] → Type ?u.15216
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.15225) → [inst : IntegralDomain R] → Type ?u.15225
Model 
R: Type u
R) (
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.15230} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.15237} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a3: {R : Type ?u.15244} → [inst : IntegralDomain R] → Model R → R
a3 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
3: ?m.15250
3)
  (
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.15267} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.15271} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a6: {R : Type ?u.15275} → [inst : IntegralDomain R] → Model R → R
a6 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
6: ?m.15279
6) : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.15289} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.15293} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
b6: {R : Type ?u.15297} → [inst : IntegralDomain R] → Model R → R
b6 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
6: ?m.15304
6 := by
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (b6 e) ≥ ofN 6simp only [
b6: {R : Type ?u.15326} → [inst : IntegralDomain R] → Model R → R
b6]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (e.a3 * e.a3 + 4 * e.a6) ≥ ofN 6
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (b6 e) ≥ ofN 6apply 
val_add_ge_of_ge: ∀ {R : Type ?u.15485} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha
SurjVal.v evr.valtn (e.a3 * e.a3) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb
SurjVal.v evr.valtn (4 * e.a6) ≥ ofN 6
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (b6 e) ≥ ofN 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha
SurjVal.v evr.valtn (e.a3 * e.a3) ≥ ofN 6 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha
SurjVal.v evr.valtn (e.a3 * e.a3) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb
SurjVal.v evr.valtn (4 * e.a6) ≥ ofN 6apply 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.15519} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.15525} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (b6 e) ≥ ofN 6.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb
SurjVal.v evr.valtn (4 * e.a6) ≥ ofN 6 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb
SurjVal.v evr.valtn (4 * e.a6) ≥ ofN 6apply 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.15549} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.15554} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6
Goals accomplished! 🐙

lemma 
val_b8_of_val_ai: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 → SurjVal.v evr.valtn e.a6 ≥ ofN 6 → SurjVal.v evr.valtn (b8 e) ≥ ofN 8
val_b8_of_val_ai (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.15584} → R → [inst : IntegralDomain R] → Type ?u.15584
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.15593) → [inst : IntegralDomain R] → Type ?u.15593
Model 
R: Type u
R) (
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.15598} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.15605} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a1: {R : Type ?u.15612} → [inst : IntegralDomain R] → Model R → R
a1 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.15618
1)
  (
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.15636} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.15640} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a2: {R : Type ?u.15644} → [inst : IntegralDomain R] → Model R → R
a2 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
2: ?m.15648
2) (
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.15658} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.15662} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a3: {R : Type ?u.15666} → [inst : IntegralDomain R] → Model R → R
a3 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
3: ?m.15670
3) (
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.15680} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.15684} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a4: {R : Type ?u.15688} → [inst : IntegralDomain R] → Model R → R
a4 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
4: ?m.15692
4)
  (
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.15702} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.15706} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a6: {R : Type ?u.15710} → [inst : IntegralDomain R] → Model R → R
a6 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
6: ?m.15714
6) : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.15724} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.15728} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
b8: {R : Type ?u.15732} → [inst : IntegralDomain R] → Model R → R
b8 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
8: ?m.15739
8 := by
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (b8 e) ≥ ofN 8simp only [
b8: {R : Type ?u.15770} → [inst : IntegralDomain R] → Model R → R
b8, 
sub_eq_add_neg: ∀ {G : Type ?u.15775} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_neg]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn
    (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 + -(e.a4 * e.a4)) ≥   ofN 8
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (b8 e) ≥ ofN 8apply 
val_add_ge_of_ge: ∀ {R : Type ?u.16128} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha
SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb
SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN 8
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (b8 e) ≥ ofN 8.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha
SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN 8 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha
SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb
SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN 8apply 
val_add_ge_of_ge: ∀ {R : Type ?u.16164} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha
SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.hb
SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN 8
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha
SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb
SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN 8.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha
SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN 8 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha
SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.hb
SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN 8apply 
val_add_ge_of_ge: ∀ {R : Type ?u.16188} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha
SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hb
SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 8
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha
SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.hb
SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN 8.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha
SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN 8 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha
SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hb
SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 8apply 
val_add_ge_of_ge: ∀ {R : Type ?u.16212} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.ha
SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.hb
SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN 8
        
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha
SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hb
SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 8.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.ha
SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN 8 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.ha
SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.hb
SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN 8apply 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.16236} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
_: SurjVal ?m.16241
_ (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.16245} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
_: SurjVal ?m.16250
_ 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1) 
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6
Goals accomplished! 🐙
        
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha
SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hb
SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 8.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.hb
SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN 8 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.hb
SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN 8rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.hb
SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN 8
val_neg: ∀ {R : Type ?u.16286} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.hb
SurjVal.v evr.valtn (e.a1 * e.a3 * e.a4) ≥ ofN 8]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.hb
SurjVal.v evr.valtn (e.a1 * e.a3 * e.a4) ≥ ofN 8
          
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.hb
SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN 8apply 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.16310} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
_: SurjVal ?m.16315
_ (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.16319} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
_: SurjVal ?m.16324
_ 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3) 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4
Goals accomplished! 🐙
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha
SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.hb
SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN 8.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hb
SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 8 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hb
SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 8rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hb
SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 8
mul_assoc: ∀ {G : Type ?u.16348} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assoc
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hb
SurjVal.v evr.valtn (4 * (e.a2 * e.a6)) ≥ ofN 8]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hb
SurjVal.v evr.valtn (4 * (e.a2 * e.a6)) ≥ ofN 8
        
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hb
SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 8apply 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.16486} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
_: SurjVal ?m.16490
_ (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.16494} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
_: SurjVal ?m.16499
_ 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6)
Goals accomplished! 🐙
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha
SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb
SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN 8.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.hb
SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN 8 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.hb
SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN 8apply 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.16528} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
_: SurjVal ?m.16533
_ (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.16537} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
_: SurjVal ?m.16542
_ 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3) 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (b8 e) ≥ ofN 8.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb
SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN 8 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb
SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN 8rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb
SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN 8
val_neg: ∀ {R : Type ?u.16566} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb
SurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN 8]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb
SurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN 8
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb
SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN 8apply 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.16590} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
_: SurjVal ?m.16595
_ 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4
Goals accomplished! 🐙

lemma 
discr_of_pi_scaling: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
          SurjVal.v evr.valtn e.a6 ≥ ofN 6 → discr (pi_scaling evr e) = EnatValRing.sub_val evr (discr e) 12
discr_of_pi_scaling (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.16634} → R → [inst : IntegralDomain R] → Type ?u.16634
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.16643) → [inst : IntegralDomain R] → Type ?u.16643
Model 
R: Type u
R) (
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.16648} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.16655} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a1: {R : Type ?u.16662} → [inst : IntegralDomain R] → Model R → R
a1 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.16668
1)
  (
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.16686} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.16690} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a2: {R : Type ?u.16694} → [inst : IntegralDomain R] → Model R → R
a2 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
2: ?m.16698
2) (
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.16708} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.16712} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a3: {R : Type ?u.16716} → [inst : IntegralDomain R] → Model R → R
a3 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
3: ?m.16720
3)
  (
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.16730} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.16734} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a4: {R : Type ?u.16738} → [inst : IntegralDomain R] → Model R → R
a4 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
4: ?m.16742
4) (
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.16752} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.16756} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a6: {R : Type ?u.16760} → [inst : IntegralDomain R] → Model R → R
a6 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
6: ?m.16764
6) :
  (
pi_scaling: {R : Type ?u.16774} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling 
evr: EnatValRing p
evr 
e: Model R
e).
discr: {R : Type ?u.16781} → [inst : IntegralDomain R] → Model R → R
discr = 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.16787} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: Model R
e.
discr: {R : Type ?u.16797} → [inst : IntegralDomain R] → Model R → R
discr 
12: ?m.16804
12 := by
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

discr (pi_scaling evr e) = EnatValRing.sub_val evr (discr e) 12simp only [
discr: {R : Type ?u.16827} → [inst : IntegralDomain R] → Model R → R
discr, 
b2_of_pi_scaling: ∀ {R : Type ?u.16842} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 → b2 (pi_scaling evr e) = EnatValRing.sub_val evr (b2 e) 2
b2_of_pi_scaling 
evr: EnatValRing p
evr 
e: Model R
e 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2, 
b4_of_pi_scaling: ∀ {R : Type ?u.16862} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
      SurjVal.v evr.valtn e.a4 ≥ ofN 4 → b4 (pi_scaling evr e) = EnatValRing.sub_val evr (b4 e) 4
b4_of_pi_scaling 
evr: EnatValRing p
evr 
e: Model R
e 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4,
    
b6_of_pi_scaling: ∀ {R : Type ?u.16877} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
    SurjVal.v evr.valtn e.a6 ≥ ofN 6 → b6 (pi_scaling evr e) = EnatValRing.sub_val evr (b6 e) 6
b6_of_pi_scaling 
evr: EnatValRing p
evr 
e: Model R
e 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6, 
b8_of_pi_scaling: ∀ {R : Type ?u.16892} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
          SurjVal.v evr.valtn e.a6 ≥ ofN 6 → b8 (pi_scaling evr e) = EnatValRing.sub_val evr (b8 e) 8
b8_of_pi_scaling 
evr: EnatValRing p
evr 
e: Model R
e 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

-EnatValRing.sub_val evr (b2 e) 2 * EnatValRing.sub_val evr (b2 e) 2 * EnatValRing.sub_val evr (b8 e) 8 -         8 * EnatValRing.sub_val evr (b4 e) 4 ^ 3 -       27 * EnatValRing.sub_val evr (b6 e) 6 * EnatValRing.sub_val evr (b6 e) 6 +     9 * EnatValRing.sub_val evr (b2 e) 2 * EnatValRing.sub_val evr (b4 e) 4 * EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e + 9 * b2 e * b4 e * b6 e) 12
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

discr (pi_scaling evr e) = EnatValRing.sub_val evr (discr e) 12exact (
pi_scaling_of_discr: ∀ {R : Type ?u.17190} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn (b2 e) ≥ ofN 2 →
    SurjVal.v evr.valtn (b4 e) ≥ ofN 4 →
      SurjVal.v evr.valtn (b6 e) ≥ ofN 6 →
        SurjVal.v evr.valtn (b8 e) ≥ ofN 8 →
          EnatValRing.sub_val evr (discr e) 12 =             -EnatValRing.sub_val evr (b2 e) 2 * EnatValRing.sub_val evr (b2 e) 2 * EnatValRing.sub_val evr (b8 e) 8 -                   8 * EnatValRing.sub_val evr (b4 e) 4 ^ 3 -                 27 * EnatValRing.sub_val evr (b6 e) 6 * EnatValRing.sub_val evr (b6 e) 6 +               9 * EnatValRing.sub_val evr (b2 e) 2 * EnatValRing.sub_val evr (b4 e) 4 * EnatValRing.sub_val evr (b6 e) 6
pi_scaling_of_discr 
evr: EnatValRing p
evr 
e: Model R
e (
val_b2_of_val_a12: ∀ {R : Type ?u.17197} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 → SurjVal.v evr.valtn e.a2 ≥ ofN 2 → SurjVal.v evr.valtn (b2 e) ≥ ofN 2
val_b2_of_val_a12 
evr: EnatValRing p
evr 
e: Model R
e 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2)
    (
val_b4_of_val_a134: ∀ {R : Type ?u.17208} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a3 ≥ ofN 3 → SurjVal.v evr.valtn e.a4 ≥ ofN 4 → SurjVal.v evr.valtn (b4 e) ≥ ofN 4
val_b4_of_val_a134 
evr: EnatValRing p
evr 
e: Model R
e 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4) (
val_b6_of_val_a36: ∀ {R : Type ?u.17215} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a3 ≥ ofN 3 → SurjVal.v evr.valtn e.a6 ≥ ofN 6 → SurjVal.v evr.valtn (b6 e) ≥ ofN 6
val_b6_of_val_a36 
evr: EnatValRing p
evr 
e: Model R
e 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6)
    (
val_b8_of_val_ai: ∀ {R : Type ?u.17222} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 → SurjVal.v evr.valtn e.a6 ≥ ofN 6 → SurjVal.v evr.valtn (b8 e) ≥ ofN 8
val_b8_of_val_ai 
evr: EnatValRing p
evr 
e: Model R
e 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6)).
symm: ∀ {α : Sort ?u.17229} {a b : α}, a = b → b = a
symm
Goals accomplished! 🐙

lemma 
val_discr_of_val_ai: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 → SurjVal.v evr.valtn e.a6 ≥ ofN 6 → SurjVal.v evr.valtn (discr e) ≥ ofN 12
val_discr_of_val_ai (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.17281} → R → [inst : IntegralDomain R] → Type ?u.17281
EnatValRing 
p: R
p) (
e: Model R
e : 
Model: (R : Type ?u.17290) → [inst : IntegralDomain R] → Type ?u.17290
Model 
R: Type u
R) (
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.17295} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.17302} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a1: {R : Type ?u.17309} → [inst : IntegralDomain R] → Model R → R
a1 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.17315
1)
  (
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.17333} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.17337} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a2: {R : Type ?u.17341} → [inst : IntegralDomain R] → Model R → R
a2 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
2: ?m.17345
2) (
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.17355} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.17359} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a3: {R : Type ?u.17363} → [inst : IntegralDomain R] → Model R → R
a3 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
3: ?m.17367
3) (
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.17377} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.17381} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a4: {R : Type ?u.17385} → [inst : IntegralDomain R] → Model R → R
a4 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
4: ?m.17389
4)
  (
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.17399} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.17403} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
a6: {R : Type ?u.17407} → [inst : IntegralDomain R] → Model R → R
a6 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
6: ?m.17411
6) : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.17421} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.17425} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: Model R
e.
discr: {R : Type ?u.17429} → [inst : IntegralDomain R] → Model R → R
discr ≥ 
ofN: ℕ → ℕ∪∞
ofN 
12: ?m.17436
12 := by
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (discr e) ≥ ofN 12have 
hb2: ?m.17461
hb2 := 
val_b2_of_val_a12: ∀ {R : Type ?u.17462} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 → SurjVal.v evr.valtn e.a2 ≥ ofN 2 → SurjVal.v evr.valtn (b2 e) ≥ ofN 2
val_b2_of_val_a12 
evr: EnatValRing p
evr 
e: Model R
e 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

SurjVal.v evr.valtn (discr e) ≥ ofN 12
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (discr e) ≥ ofN 12have 
hb4: ?m.17478
hb4 := 
val_b4_of_val_a134: ∀ {R : Type ?u.17479} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a3 ≥ ofN 3 → SurjVal.v evr.valtn e.a4 ≥ ofN 4 → SurjVal.v evr.valtn (b4 e) ≥ ofN 4
val_b4_of_val_a134 
evr: EnatValRing p
evr 
e: Model R
e 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

SurjVal.v evr.valtn (discr e) ≥ ofN 12
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (discr e) ≥ ofN 12have 
hb6: ?m.17491
hb6 := 
val_b6_of_val_a36: ∀ {R : Type ?u.17492} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a3 ≥ ofN 3 → SurjVal.v evr.valtn e.a6 ≥ ofN 6 → SurjVal.v evr.valtn (b6 e) ≥ ofN 6
val_b6_of_val_a36 
evr: EnatValRing p
evr 
e: Model R
e 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

SurjVal.v evr.valtn (discr e) ≥ ofN 12
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (discr e) ≥ ofN 12have 
hb8: ?m.17504
hb8 := 
val_b8_of_val_ai: ∀ {R : Type ?u.17505} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 → SurjVal.v evr.valtn e.a6 ≥ ofN 6 → SurjVal.v evr.valtn (b8 e) ≥ ofN 8
val_b8_of_val_ai 
evr: EnatValRing p
evr 
e: Model R
e 
h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 
h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 
h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 
h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 
h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (discr e) ≥ ofN 12
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (discr e) ≥ ofN 12simp only [
discr: {R : Type ?u.17523} → [inst : IntegralDomain R] → Model R → R
discr, 
sub_eq_add_neg: ∀ {G : Type ?u.17537} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_neg]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3) + -(27 * b6 e * b6 e) + 9 * b2 e * b4 e * b6 e) ≥ ofN 12
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (discr e) ≥ ofN 12apply 
val_add_ge_of_ge: ∀ {R : Type ?u.17954} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3) + -(27 * b6 e * b6 e)) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hb
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (discr e) ≥ ofN 12.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3) + -(27 * b6 e * b6 e)) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3) + -(27 * b6 e * b6 e)) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hb
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12apply 
val_add_ge_of_ge: ∀ {R : Type ?u.17984} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3)) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hb
SurjVal.v evr.valtn (-(27 * b6 e * b6 e)) ≥ ofN 12
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3) + -(27 * b6 e * b6 e)) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hb
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3)) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3)) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hb
SurjVal.v evr.valtn (-(27 * b6 e * b6 e)) ≥ ofN 12apply 
val_add_ge_of_ge: ∀ {R : Type ?u.18010} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hb
SurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3)) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hb
SurjVal.v evr.valtn (-(27 * b6 e * b6 e)) ≥ ofN 12.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hb
SurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12←
neg_mul_left: ∀ {R : Type ?u.18034} [inst : Ring R] (a b : R), -(a * b) = -a * b
neg_mul_left,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.ha
SurjVal.v evr.valtn (-(b2 e * b2 e) * b8 e) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12←
neg_mul_left: ∀ {R : Type ?u.18121} [inst : Ring R] (a b : R), -(a * b) = -a * b
neg_mul_left,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.ha
SurjVal.v evr.valtn (-(b2 e * b2 e * b8 e)) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
val_neg: ∀ {R : Type ?u.18159} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.ha
SurjVal.v evr.valtn (b2 e * b2 e * b8 e) ≥ ofN 12]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.ha
SurjVal.v evr.valtn (b2 e * b2 e * b8 e) ≥ ofN 12
          
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hb
SurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12apply 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.18183} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
_: SurjVal ?m.18188
_ (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.18192} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
_: SurjVal ?m.18197
_ 
hb2: ?m.17461
hb2 
hb2: ?m.17461
hb2) 
hb8: ?m.17504
hb8
Goals accomplished! 🐙
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3)) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hb
SurjVal.v evr.valtn (-(27 * b6 e * b6 e)) ≥ ofN 12.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hb
SurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hb
SurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hb
SurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12
val_neg: ∀ {R : Type ?u.18233} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hb
SurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hb
SurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12
pow_succ': ∀ {M : Type ?u.18249} [inst : Monoid M] (a : M) (n : ℕ), a ^ (n + 1) = a ^ n * a
pow_succ',
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hb
SurjVal.v evr.valtn (8 * (b4 e ^ 2 * b4 e)) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hb
SurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12
pow_succ': ∀ {M : Type ?u.18366} [inst : Monoid M] (a : M) (n : ℕ), a ^ (n + 1) = a ^ n * a
pow_succ',
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hb
SurjVal.v evr.valtn (8 * (b4 e ^ 1 * b4 e * b4 e)) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hb
SurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12
pow_one: ∀ {M : Type ?u.18440} [inst : Monoid M] (a : M), a ^ 1 = a
pow_one
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hb
SurjVal.v evr.valtn (8 * (b4 e * b4 e * b4 e)) ≥ ofN 12]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hb
SurjVal.v evr.valtn (8 * (b4 e * b4 e * b4 e)) ≥ ofN 12
        
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hb
SurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12apply 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.18521} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
_: SurjVal ?m.18525
_ (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.18529} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
_: SurjVal ?m.18534
_ (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.18547} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
_: SurjVal ?m.18552
_ 
hb4: ?m.17478
hb4 
hb4: ?m.17478
hb4) 
hb4: ?m.17478
hb4)
Goals accomplished! 🐙
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha
SurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3) + -(27 * b6 e * b6 e)) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hb
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hb
SurjVal.v evr.valtn (-(27 * b6 e * b6 e)) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hb
SurjVal.v evr.valtn (-(27 * b6 e * b6 e)) ≥ ofN 12rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hb
SurjVal.v evr.valtn (-(27 * b6 e * b6 e)) ≥ ofN 12
val_neg: ∀ {R : Type ?u.18584} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hb
SurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hb
SurjVal.v evr.valtn (-(27 * b6 e * b6 e)) ≥ ofN 12
mul_assoc: ∀ {G : Type ?u.18599} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assoc
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hb
SurjVal.v evr.valtn (27 * (b6 e * b6 e)) ≥ ofN 12]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hb
SurjVal.v evr.valtn (27 * (b6 e * b6 e)) ≥ ofN 12
      
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hb
SurjVal.v evr.valtn (-(27 * b6 e * b6 e)) ≥ ofN 12apply 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.18732} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
_: SurjVal ?m.18736
_ (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.18740} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
_: SurjVal ?m.18745
_ 
hb6: ?m.17491
hb6 
hb6: ?m.17491
hb6)
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (discr e) ≥ ofN 12.
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hb
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hb
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hb
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
mul_assoc: ∀ {G : Type ?u.18766} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assoc,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hb
SurjVal.v evr.valtn (9 * b2 e * (b4 e * b6 e)) ≥ ofN 12 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hb
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
mul_assoc: ∀ {G : Type ?u.18850} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assoc
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hb
SurjVal.v evr.valtn (9 * (b2 e * (b4 e * b6 e))) ≥ ofN 12]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hb
SurjVal.v evr.valtn (9 * (b2 e * (b4 e * b6 e))) ≥ ofN 12
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hb
SurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12apply 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.18943} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
_: SurjVal ?m.18947
_ (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.18951} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
_: SurjVal ?m.18956
_ 
hb2: ?m.17461
hb2 (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.18974} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
_: SurjVal ?m.18979
_ 
hb4: ?m.17478
hb4 
hb6: ?m.17491
hb6))
Goals accomplished! 🐙

end Model


namespace ValidModel

def 
pi_scaling: {p : R} →
  (evr : EnatValRing p) →
    (e : ValidModel R) →
      SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1 →
        SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2 →
          SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3 →
            SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4 → SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6 → ValidModel R
pi_scaling (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.19032} → R → [inst : IntegralDomain R] → Type ?u.19032
EnatValRing 
p: ?m.19029
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.19040) → [inst : IntegralDomain R] → Type ?u.19040
ValidModel 
R: Type u
R) (
h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1
h1 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.19049} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.19062} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a1: {R : Type ?u.19077} → [inst : IntegralDomain R] → Model R → R
a1 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.19083
1)
  (
h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2
h2 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.19101} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.19108} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a2: {R : Type ?u.19118} → [inst : IntegralDomain R] → Model R → R
a2 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
2: ?m.19122
2) (
h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3
h3 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.19132} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.19136} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a3: {R : Type ?u.19143} → [inst : IntegralDomain R] → Model R → R
a3 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
3: ?m.19147
3) (
h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4
h4 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.19157} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.19161} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a4: {R : Type ?u.19168} → [inst : IntegralDomain R] → Model R → R
a4 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
4: ?m.19172
4)
  (
h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6
h6 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.19182} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.19186} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a6: {R : Type ?u.19193} → [inst : IntegralDomain R] → Model R → R
a6 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
6: ?m.19197
6) : 
ValidModel: (R : Type ?u.19206) → [inst : IntegralDomain R] → Type ?u.19206
ValidModel 
R: Type u
R := {
  toModel := 
Model.pi_scaling: {R : Type ?u.19220} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
Model.pi_scaling 
evr: EnatValRing p
evr 
e: ValidModel R
e.
toModel: {R : Type ?u.19224} → [inst : IntegralDomain R] → ValidModel R → Model R
toModel,
  discr_not_zero := by
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

Model.discr (Model.pi_scaling evr e.toModel) ≠ 0rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

Model.discr (Model.pi_scaling evr e.toModel) ≠ 0
Model.discr_of_pi_scaling: ∀ {R : Type ?u.19229} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
          SurjVal.v evr.valtn e.a6 ≥ ofN 6 →
            Model.discr (Model.pi_scaling evr e) = EnatValRing.sub_val evr (Model.discr e) 12
Model.discr_of_pi_scaling 
evr: EnatValRing p
evr 
e: ValidModel R
e.
toModel: {R : Type ?u.19236} → [inst : IntegralDomain R] → ValidModel R → Model R
toModel 
h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1
h1 
h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2
h2 
h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3
h3 
h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4
h4 
h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6
h6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

EnatValRing.sub_val evr (Model.discr e.toModel) 12 ≠ 0]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

EnatValRing.sub_val evr (Model.discr e.toModel) 12 ≠ 0
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

Model.discr (Model.pi_scaling evr e.toModel) ≠ 0intro 
H: EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0
H
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

H: EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0

False
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

Model.discr (Model.pi_scaling evr e.toModel) ≠ 0have 
H': ?m.19264
H' := 
let_value_eq: ∀ {α : Sort ?u.19266} {β : Sort ?u.19265} {a₁ a₂ : α} (b : α → β),
  a₁ = a₂ →
    (let x := a₁;
      b x) =       let x := a₂;
      b x
let_value_eq (fun (
x: R
x:
R: Type u
R) => 
p: R
p ^ 
12: ?m.19280
12 * 
x: R
x) 
H: EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0
H
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

H: EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0

H': (let x := EnatValRing.sub_val evr (Model.discr e.toModel) 12;
  (fun x => p ^ 12 * x) x) =   let x := 0;
  (fun x => p ^ 12 * x) x

False
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

Model.discr (Model.pi_scaling evr e.toModel) ≠ 0simp only [
mul_zero: ∀ {M₀ : Type ?u.19608} [self : MonoidWithZero M₀] (a : M₀), a * 0 = 0
mul_zero] at 
H': ?m.19264
H'
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

H: EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0

H': p ^ 12 * EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0

False
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

Model.discr (Model.pi_scaling evr e.toModel) ≠ 0rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

H: EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0

H': p ^ 12 * EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0

False←
evr: EnatValRing p
evr.
factor_p_of_le_val: ∀ {R : Type ?u.19836} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → x = p ^ n * EnatValRing.sub_val evr x n
factor_p_of_le_val (
Model.val_discr_of_val_ai: ∀ {R : Type ?u.19849} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
          SurjVal.v evr.valtn e.a6 ≥ ofN 6 → SurjVal.v evr.valtn (Model.discr e) ≥ ofN 12
Model.val_discr_of_val_ai 
evr: EnatValRing p
evr 
e: ValidModel R
e.
toModel: {R : Type ?u.19856} → [inst : IntegralDomain R] → ValidModel R → Model R
toModel 
h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1
h1 
h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2
h2 
h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3
h3 
h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4
h4 
h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6
h6)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

H: EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0

H': Model.discr e.toModel = 0

False]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

H: EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0

H': Model.discr e.toModel = 0

False at 
H': p ^ 12 * EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0
H'
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

H: EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0

H': Model.discr e.toModel = 0

False
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

Model.discr (Model.pi_scaling evr e.toModel) ≠ 0apply 
e: ValidModel R
e.
discr_not_zero: ∀ {R : Type ?u.19882} [inst : IntegralDomain R] (self : ValidModel R), Model.discr self.toModel ≠ 0
discr_not_zero 
H': Model.discr e.toModel = 0
H'
Goals accomplished! 🐙 }

lemma 
toModel_pi_scaling: ∀ {p : R} (evr : EnatValRing p) (e : ValidModel R) (h1 : SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1)
  (h2 : SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2) (h3 : SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3)
  (h4 : SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4) (h6 : SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6),
  (pi_scaling evr e h1 h2 h3 h4 h6).toModel = Model.pi_scaling evr e.toModel
toModel_pi_scaling (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.20002} → R → [inst : IntegralDomain R] → Type ?u.20002
EnatValRing 
p: ?m.19999
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.20010) → [inst : IntegralDomain R] → Type ?u.20010
ValidModel 
R: Type u
R) (
h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1
h1 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.20019} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.20032} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a1: {R : Type ?u.20047} → [inst : IntegralDomain R] → Model R → R
a1 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.20053
1)
  (
h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2
h2 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.20071} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.20078} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a2: {R : Type ?u.20088} → [inst : IntegralDomain R] → Model R → R
a2 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
2: ?m.20092
2) (
h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3
h3 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.20102} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.20106} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a3: {R : Type ?u.20113} → [inst : IntegralDomain R] → Model R → R
a3 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
3: ?m.20117
3) (
h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4
h4 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.20127} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.20131} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a4: {R : Type ?u.20138} → [inst : IntegralDomain R] → Model R → R
a4 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
4: ?m.20142
4)
  (
h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6
h6 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.20152} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.20156} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a6: {R : Type ?u.20163} → [inst : IntegralDomain R] → Model R → R
a6 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
6: ?m.20167
6) :
  (
e: ValidModel R
e.
pi_scaling: {R : Type ?u.20177} →
  [inst : IntegralDomain R] →
    {p : R} →
      (evr : EnatValRing p) →
        (e : ValidModel R) →
          SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1 →
            SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2 →
              SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3 →
                SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4 → SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6 → ValidModel R
pi_scaling 
evr: EnatValRing p
evr 
h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1
h1 
h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2
h2 
h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3
h3 
h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4
h4 
h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6
h6).
toModel: {R : Type ?u.20198} → [inst : IntegralDomain R] → ValidModel R → Model R
toModel = 
e: ValidModel R
e.
toModel: {R : Type ?u.20201} → [inst : IntegralDomain R] → ValidModel R → Model R
toModel.
pi_scaling: {R : Type ?u.20204} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling 
evr: EnatValRing p
evr := 
rfl: ∀ {α : Sort ?u.20226} {a : α}, a = a
rfl

def 
val_discr_to_nat: {R : Type u} → [inst : IntegralDomain R] → {p : R} → SurjVal p → ValidModel R → ℕ
val_discr_to_nat {
p: R
p : 
R: Type u
R} (
valp: SurjVal p
valp : 
SurjVal: {R : Type ?u.20418} → R → [inst : IntegralDomain R] → Type ?u.20418
SurjVal 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.20427) → [inst : IntegralDomain R] → Type ?u.20427
ValidModel 
R: Type u
R) : 
ℕ: Type
ℕ :=
nat_of_val: {R : Type ?u.20435} → [inst : IntegralDomain R] → {p : R} → SurjVal p → {a : R} → a ≠ 0 → ℕ
nat_of_val 
valp: SurjVal p
valp 
e: ValidModel R
e.
discr_not_zero: ∀ {R : Type ?u.20446} [inst : IntegralDomain R] (self : ValidModel R), Model.discr self.toModel ≠ 0
discr_not_zero

@[simp]
lemma 
iso_rst_val_discr_to_nat: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (valp : SurjVal p) (r s t : R) (e : ValidModel R),
  val_discr_to_nat valp (rst_iso r s t e) = val_discr_to_nat valp e
iso_rst_val_discr_to_nat {
p: R
p : 
R: Type u
R} (
valp: SurjVal p
valp : 
SurjVal: {R : Type ?u.20507} → R → [inst : IntegralDomain R] → Type ?u.20507
SurjVal 
p: R
p) (
r: R
r 
s: R
s 
t: R
t : 
R: Type u
R) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.20522) → [inst : IntegralDomain R] → Type ?u.20522
ValidModel 
R: Type u
R) :
  
val_discr_to_nat: {R : Type ?u.20527} → [inst : IntegralDomain R] → {p : R} → SurjVal p → ValidModel R → ℕ
val_discr_to_nat 
valp: SurjVal p
valp (
rst_iso: {R : Type ?u.20537} → [inst : IntegralDomain R] → R → R → R → ValidModel R → ValidModel R
rst_iso 
r: R
r 
s: R
s 
t: R
t 
e: ValidModel R
e) = 
val_discr_to_nat: {R : Type ?u.20542} → [inst : IntegralDomain R] → {p : R} → SurjVal p → ValidModel R → ℕ
val_discr_to_nat 
valp: SurjVal p
valp 
e: ValidModel R
e :=
by 
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

r, s, t: R

e: ValidModel R

val_discr_to_nat valp (rst_iso r s t e) = val_discr_to_nat valp esimp [
val_discr_to_nat: {R : Type ?u.20559} → [inst : IntegralDomain R] → {p : R} → SurjVal p → ValidModel R → ℕ
val_discr_to_nat, 
nat_of_val: {R : Type ?u.20561} → [inst : IntegralDomain R] → {p : R} → SurjVal p → {a : R} → a ≠ 0 → ℕ
nat_of_val, 
ValidModel.rst_iso: {R : Type ?u.20563} → [inst : IntegralDomain R] → R → R → R → ValidModel R → ValidModel R
ValidModel.rst_iso, 
Model.rst_discr: ∀ {R : Type ?u.20565} [inst : IntegralDomain R] (r s t : R) (e : Model R),
  Model.discr (Model.rst_iso r s t e) = Model.discr e
Model.rst_discr]
Goals accomplished! 🐙


lemma 
ofN_val_discr_to_nat: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (valp : SurjVal p) (e : ValidModel R),
  ofN (val_discr_to_nat valp e) = SurjVal.v valp (Model.discr e.toModel)
ofN_val_discr_to_nat {
p: R
p : 
R: Type u
R} (
valp: SurjVal p
valp : 
SurjVal: {R : Type ?u.20944} → R → [inst : IntegralDomain R] → Type ?u.20944
SurjVal 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.20953) → [inst : IntegralDomain R] → Type ?u.20953
ValidModel 
R: Type u
R) :
  
ofN: ℕ → ℕ∪∞
ofN (
val_discr_to_nat: {R : Type ?u.20958} → [inst : IntegralDomain R] → {p : R} → SurjVal p → ValidModel R → ℕ
val_discr_to_nat 
valp: SurjVal p
valp 
e: ValidModel R
e) = 
valp: SurjVal p
valp.
v: {R : Type ?u.20970} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
discr: {R : Type ?u.20977} → [inst : IntegralDomain R] → Model R → R
discr := by
  
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

ofN (val_discr_to_nat valp e) = SurjVal.v valp (Model.discr e.toModel)cases h : 
valp: SurjVal p
valp.
v: {R : Type ?u.20992} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
discr: {R : Type ?u.20999} → [inst : IntegralDomain R] → Model R → R
discr with
  
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

x✝: ℕ∪∞

h: SurjVal.v valp (Model.discr e.toModel) = x✝

ofN (val_discr_to_nat valp e) = x✝| 
ofN: ℕ → ℕ∪∞
ofN 
n: ℕ
n =>
    
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

n: ℕ

h: SurjVal.v valp (Model.discr e.toModel) = ofN n

ofN
ofN (val_discr_to_nat valp e) = ofN nrw [
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

n: ℕ

h: SurjVal.v valp (Model.discr e.toModel) = ofN n

ofN
ofN (val_discr_to_nat valp e) = ofN n
val_discr_to_nat: {R : Type ?u.21087} → [inst : IntegralDomain R] → {p : R} → SurjVal p → ValidModel R → ℕ
val_discr_to_nat,
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

n: ℕ

h: SurjVal.v valp (Model.discr e.toModel) = ofN n

ofN
ofN (nat_of_val valp (_ : Model.discr e.toModel ≠ 0)) = ofN n 
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

n: ℕ

h: SurjVal.v valp (Model.discr e.toModel) = ofN n

ofN
ofN (val_discr_to_nat valp e) = ofN n
nat_of_val: {R : Type ?u.21126} → [inst : IntegralDomain R] → {p : R} → SurjVal p → {a : R} → a ≠ 0 → ℕ
nat_of_val,
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

n: ℕ

h: SurjVal.v valp (Model.discr e.toModel) = ofN n

ofN
ofN (to_nat (_ : ¬SurjVal.v valp (Model.discr e.toModel) = ∞)) = ofN n 
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

n: ℕ

h: SurjVal.v valp (Model.discr e.toModel) = ofN n

ofN
ofN (val_discr_to_nat valp e) = ofN n
ofN_to_nat_eq_self: ∀ {a : ℕ∪∞} (h : a ≠ ∞), ofN (to_nat h) = a
ofN_to_nat_eq_self
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

n: ℕ

h: SurjVal.v valp (Model.discr e.toModel) = ofN n

ofN
SurjVal.v valp (Model.discr e.toModel) = ofN n]
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

n: ℕ

h: SurjVal.v valp (Model.discr e.toModel) = ofN n

ofN
SurjVal.v valp (Model.discr e.toModel) = ofN n
    
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

n: ℕ

h: SurjVal.v valp (Model.discr e.toModel) = ofN n

ofN
ofN (val_discr_to_nat valp e) = ofN nassumption
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

x✝: ℕ∪∞

h: SurjVal.v valp (Model.discr e.toModel) = x✝

ofN (val_discr_to_nat valp e) = x✝| 
top: ℕ∪∞
top =>
    
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

h: SurjVal.v valp (Model.discr e.toModel) = ∞

top
ofN (val_discr_to_nat valp e) = ∞exfalso
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

h: SurjVal.v valp (Model.discr e.toModel) = ∞

top.h
False
    
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

h: SurjVal.v valp (Model.discr e.toModel) = ∞

top
ofN (val_discr_to_nat valp e) = ∞rw [
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

h: SurjVal.v valp (Model.discr e.toModel) = ∞

top.h
False
valp: SurjVal p
valp.
v_eq_top_iff_zero: ∀ {R : Type ?u.21162} {p : R} [inst : IntegralDomain R] (self : SurjVal p) (a : R), SurjVal.v self a = ∞ ↔ a = 0
v_eq_top_iff_zero
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

h: Model.discr e.toModel = 0

top.h
False]
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

h: Model.discr e.toModel = 0

top.h
False at 
h: SurjVal.v valp (Model.discr e.toModel) = ∞
h
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

h: Model.discr e.toModel = 0

top.h
False
    
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

h: SurjVal.v valp (Model.discr e.toModel) = ∞

top
ofN (val_discr_to_nat valp e) = ∞exact 
e: ValidModel R
e.
discr_not_zero: ∀ {R : Type ?u.21193} [inst : IntegralDomain R] (self : ValidModel R), Model.discr self.toModel ≠ 0
discr_not_zero 
h: Model.discr e.toModel = 0
h
Goals accomplished! 🐙

lemma 
pi_scaling_val_discr_to_nat: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : ValidModel R)
  (h1 : SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1) (h2 : SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2)
  (h3 : SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3) (h4 : SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4)
  (h6 : SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6),
  val_discr_to_nat evr.valtn (pi_scaling evr e h1 h2 h3 h4 h6) = val_discr_to_nat evr.valtn e - 12
pi_scaling_val_discr_to_nat {
p: R
p : 
R: Type u
R} (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.21227} → R → [inst : IntegralDomain R] → Type ?u.21227
EnatValRing 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.21236) → [inst : IntegralDomain R] → Type ?u.21236
ValidModel 
R: Type u
R)
  (
h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1
h1 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.21241} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.21248} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a1: {R : Type ?u.21260} → [inst : IntegralDomain R] → Model R → R
a1 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.21266
1) (
h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2
h2 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.21284} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.21288} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a2: {R : Type ?u.21295} → [inst : IntegralDomain R] → Model R → R
a2 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
2: ?m.21299
2)
  (
h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3
h3 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.21309} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.21313} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a3: {R : Type ?u.21320} → [inst : IntegralDomain R] → Model R → R
a3 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
3: ?m.21324
3) (
h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4
h4 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.21334} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.21338} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a4: {R : Type ?u.21345} → [inst : IntegralDomain R] → Model R → R
a4 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
4: ?m.21349
4) (
h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6
h6 : 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.21359} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.21363} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a6: {R : Type ?u.21370} → [inst : IntegralDomain R] → Model R → R
a6 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
6: ?m.21374
6) :
  
val_discr_to_nat: {R : Type ?u.21384} → [inst : IntegralDomain R] → {p : R} → SurjVal p → ValidModel R → ℕ
val_discr_to_nat 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.21388} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (
pi_scaling: {R : Type ?u.21395} →
  [inst : IntegralDomain R] →
    {p : R} →
      (evr : EnatValRing p) →
        (e : ValidModel R) →
          SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1 →
            SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2 →
              SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3 →
                SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4 → SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6 → ValidModel R
pi_scaling 
evr: EnatValRing p
evr 
e: ValidModel R
e 
h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1
h1 
h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2
h2 
h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3
h3 
h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4
h4 
h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6
h6) = 
val_discr_to_nat: {R : Type ?u.21406} → [inst : IntegralDomain R] → {p : R} → SurjVal p → ValidModel R → ℕ
val_discr_to_nat 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.21410} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 
e: ValidModel R
e - 
12: ?m.21418
12 :=
by
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

val_discr_to_nat evr.valtn (pi_scaling evr e h1 h2 h3 h4 h6) = val_discr_to_nat evr.valtn e - 12rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

val_discr_to_nat evr.valtn (pi_scaling evr e h1 h2 h3 h4 h6) = val_discr_to_nat evr.valtn e - 12
Enat.eq_ofN: ∀ (m n : ℕ), m = n ↔ ofN m = ofN n
Enat.eq_ofN,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

ofN (val_discr_to_nat evr.valtn (pi_scaling evr e h1 h2 h3 h4 h6)) = ofN (val_discr_to_nat evr.valtn e - 12) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

val_discr_to_nat evr.valtn (pi_scaling evr e h1 h2 h3 h4 h6) = val_discr_to_nat evr.valtn e - 12
ofN_val_discr_to_nat: ∀ {R : Type ?u.21493} [inst : IntegralDomain R] {p : R} (valp : SurjVal p) (e : ValidModel R),
  ofN (val_discr_to_nat valp e) = SurjVal.v valp (Model.discr e.toModel)
ofN_val_discr_to_nat,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

SurjVal.v evr.valtn (Model.discr (pi_scaling evr e h1 h2 h3 h4 h6).toModel) = ofN (val_discr_to_nat evr.valtn e - 12) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

val_discr_to_nat evr.valtn (pi_scaling evr e h1 h2 h3 h4 h6) = val_discr_to_nat evr.valtn e - 12
toModel_pi_scaling: ∀ {R : Type ?u.21521} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : ValidModel R)
  (h1 : SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1) (h2 : SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2)
  (h3 : SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3) (h4 : SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4)
  (h6 : SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6),
  (pi_scaling evr e h1 h2 h3 h4 h6).toModel = Model.pi_scaling evr e.toModel
toModel_pi_scaling 
evr: EnatValRing p
evr 
e: ValidModel R
e 
h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1
h1 
h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2
h2 
h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3
h3 
h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4
h4 
h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6
h6,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

SurjVal.v evr.valtn (Model.discr (Model.pi_scaling evr e.toModel)) = ofN (val_discr_to_nat evr.valtn e - 12)
    
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

val_discr_to_nat evr.valtn (pi_scaling evr e h1 h2 h3 h4 h6) = val_discr_to_nat evr.valtn e - 12
Model.discr_of_pi_scaling: ∀ {R : Type ?u.21531} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
          SurjVal.v evr.valtn e.a6 ≥ ofN 6 →
            Model.discr (Model.pi_scaling evr e) = EnatValRing.sub_val evr (Model.discr e) 12
Model.discr_of_pi_scaling 
_: EnatValRing ?m.21534
_ 
_: Model ?m.21532
_ 
h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1
h1 
h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2
h2 
h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3
h3 
h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4
h4 
h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6
h6,
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

SurjVal.v evr.valtn (EnatValRing.sub_val evr (Model.discr e.toModel) 12) = ofN (val_discr_to_nat evr.valtn e - 12) 
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

val_discr_to_nat evr.valtn (pi_scaling evr e h1 h2 h3 h4 h6) = val_discr_to_nat evr.valtn e - 12
evr: EnatValRing p
evr.
val_sub_val_eq: ∀ {R : Type ?u.21552} [inst : IntegralDomain R] {m : ℕ} {p : R} (evr : EnatValRing p) (x : R) (n : ℕ),
  SurjVal.v evr.valtn x = ofN m → SurjVal.v evr.valtn (EnatValRing.sub_val evr x n) = ofN (m - n)
val_sub_val_eq
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

ofN (?m.21563 - 12) = ofN (val_discr_to_nat evr.valtn e - 12)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

h
SurjVal.v evr.valtn (Model.discr e.toModel) = ofN ?m.21563
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

ℕ]
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

h
SurjVal.v evr.valtn (Model.discr e.toModel) = ofN (val_discr_to_nat evr.valtn e)
  
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

val_discr_to_nat evr.valtn (pi_scaling evr e h1 h2 h3 h4 h6) = val_discr_to_nat evr.valtn e - 12rw [
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

h
SurjVal.v evr.valtn (Model.discr e.toModel) = ofN (val_discr_to_nat evr.valtn e)
ofN_val_discr_to_nat: ∀ {R : Type ?u.21595} [inst : IntegralDomain R] {p : R} (valp : SurjVal p) (e : ValidModel R),
  ofN (val_discr_to_nat valp e) = SurjVal.v valp (Model.discr e.toModel)
ofN_val_discr_to_nat
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

h
SurjVal.v evr.valtn (Model.discr e.toModel) = SurjVal.v evr.valtn (Model.discr e.toModel)]
Goals accomplished! 🐙

lemma 
v_b2_of_v_a1_a2: ∀ {p : R} (valp : SurjVal p) (e : ValidModel R),
  SurjVal.v valp e.toModel.a1 ≥ ofN 1 →
    SurjVal.v valp e.toModel.a2 = ofN 1 → SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1
v_b2_of_v_a1_a2 {
p: R
p : 
R: Type u
R} (
valp: SurjVal p
valp : 
SurjVal: {R : Type ?u.21639} → R → [inst : IntegralDomain R] → Type ?u.21639
SurjVal 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.21648) → [inst : IntegralDomain R] → Type ?u.21648
ValidModel 
R: Type u
R) (
h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1 : 
valp: SurjVal p
valp.
v: {R : Type ?u.21653} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a1: {R : Type ?u.21665} → [inst : IntegralDomain R] → Model R → R
a1 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.21671
1)
  (
h2: SurjVal.v valp e.toModel.a2 = ofN 1
h2 : 
valp: SurjVal p
valp.
v: {R : Type ?u.21689} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a2: {R : Type ?u.21696} → [inst : IntegralDomain R] → Model R → R
a2 = 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.21700
1) : 
valp: SurjVal p
valp.
v: {R : Type ?u.21706} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
b2: {R : Type ?u.21713} → [inst : IntegralDomain R] → Model R → R
b2 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.21720
1 :=
  
val_add_ge_of_ge: ∀ {R : Type ?u.21735} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge 
valp: SurjVal p
valp (
val_mul_ge_of_left_ge: ∀ {R : Type ?u.21760} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_left_ge 
valp: SurjVal p
valp 
h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1) (
val_mul_ge_of_right_ge: ∀ {R : Type ?u.21776} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
valp: SurjVal p
valp (
le_of_eq: ∀ {α : Type ?u.21786} [inst : Preorder α] {a b : α}, a = b → a ≤ b
le_of_eq 
h2: SurjVal.v valp e.toModel.a2 = ofN 1
h2.
symm: ∀ {α : Sort ?u.21797} {a b : α}, a = b → b = a
symm))

lemma 
v_b4_of_v_a1_a3_a4: ∀ {R : Type u} [inst : IntegralDomain R] {q : ℕ} {p : R} (valp : SurjVal p) (e : ValidModel R),
  SurjVal.v valp e.toModel.a1 ≥ ofN 1 →
    SurjVal.v valp e.toModel.a3 ≥ ofN q →
      SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1) → SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)
v_b4_of_v_a1_a3_a4 {
p: R
p : 
R: Type u
R} (
valp: SurjVal p
valp : 
SurjVal: {R : Type ?u.21890} → R → [inst : IntegralDomain R] → Type ?u.21890
SurjVal 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.21895) → [inst : IntegralDomain R] → Type ?u.21895
ValidModel 
R: Type u
R) (
h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1 : 
valp: SurjVal p
valp.
v: {R : Type ?u.21900} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a1: {R : Type ?u.21907} → [inst : IntegralDomain R] → Model R → R
a1 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.21911
1)
  (
h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 : 
valp: SurjVal p
valp.
v: {R : Type ?u.21918} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a3: {R : Type ?u.21925} → [inst : IntegralDomain R] → Model R → R
a3 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
q: ?m.21885
q) (
h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)
h4 : 
valp: SurjVal p
valp.
v: {R : Type ?u.21933} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a4: {R : Type ?u.21940} → [inst : IntegralDomain R] → Model R → R
a4 ≥ 
ofN: ℕ → ℕ∪∞
ofN (
q: ?m.21885
q + 
1: ?m.21947
1)) : 
valp: SurjVal p
valp.
v: {R : Type ?u.22004} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
b4: {R : Type ?u.22011} → [inst : IntegralDomain R] → Model R → R
b4 ≥ 
ofN: ℕ → ℕ∪∞
ofN (
q: ?m.21885
q + 
1: ?m.22021
1) := by
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)apply 
val_add_ge_of_ge: ∀ {R : Type ?u.22076} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge 
valp: SurjVal p
valp
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

ha
SurjVal.v valp (e.toModel.a1 * e.toModel.a3) ≥ ofN (q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

hb
SurjVal.v valp (2 * e.toModel.a4) ≥ ofN (q + 1)
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1).
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

ha
SurjVal.v valp (e.toModel.a1 * e.toModel.a3) ≥ ofN (q + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

ha
SurjVal.v valp (e.toModel.a1 * e.toModel.a3) ≥ ofN (q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

hb
SurjVal.v valp (2 * e.toModel.a4) ≥ ofN (q + 1)rw [
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

ha
SurjVal.v valp (e.toModel.a1 * e.toModel.a3) ≥ ofN (q + 1)←
add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

ha
SurjVal.v valp (e.toModel.a1 * e.toModel.a3) ≥ ofN q + ofN 1 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

ha
SurjVal.v valp (e.toModel.a1 * e.toModel.a3) ≥ ofN (q + 1)
Enat.add_comm: ∀ (a b : ℕ∪∞), a + b = b + a
Enat.add_comm
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

ha
SurjVal.v valp (e.toModel.a1 * e.toModel.a3) ≥ ofN 1 + ofN q]
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

ha
SurjVal.v valp (e.toModel.a1 * e.toModel.a3) ≥ ofN 1 + ofN q
    
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

ha
SurjVal.v valp (e.toModel.a1 * e.toModel.a3) ≥ ofN (q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

hb
SurjVal.v valp (2 * e.toModel.a4) ≥ ofN (q + 1)exact (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.22137} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
valp: SurjVal p
valp 
h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1 
h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3)
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1).
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

hb
SurjVal.v valp (2 * e.toModel.a4) ≥ ofN (q + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

hb
SurjVal.v valp (2 * e.toModel.a4) ≥ ofN (q + 1)exact (
val_mul_ge_of_right_ge: ∀ {R : Type ?u.22160} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
valp: SurjVal p
valp 
h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)
h4)
Goals accomplished! 🐙

lemma 
v_b6_of_v_a3_a6: ∀ {q : ℕ} {p : R} (valp : SurjVal p) (e : ValidModel R),
  SurjVal.v valp e.toModel.a3 ≥ ofN q →
    SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q) → SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)
v_b6_of_v_a3_a6 {
p: R
p : 
R: Type u
R} (
valp: SurjVal p
valp : 
SurjVal: {R : Type ?u.22226} → R → [inst : IntegralDomain R] → Type ?u.22226
SurjVal 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.22231) → [inst : IntegralDomain R] → Type ?u.22231
ValidModel 
R: Type u
R) (
h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 : 
valp: SurjVal p
valp.
v: {R : Type ?u.22236} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a3: {R : Type ?u.22243} → [inst : IntegralDomain R] → Model R → R
a3 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
q: ?m.22221
q)
  (
h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)
h6 : 
valp: SurjVal p
valp.
v: {R : Type ?u.22254} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a6: {R : Type ?u.22261} → [inst : IntegralDomain R] → Model R → R
a6 ≥ 
ofN: ℕ → ℕ∪∞
ofN (
2: ?m.22268
2 * 
q: ?m.22221
q)) : 
valp: SurjVal p
valp.
v: {R : Type ?u.22334} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
b6: {R : Type ?u.22341} → [inst : IntegralDomain R] → Model R → R
b6 ≥ 
ofN: ℕ → ℕ∪∞
ofN (
2: ?m.22351
2 * 
q: ?m.22221
q) := by
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)apply 
val_add_ge_of_ge: ∀ {R : Type ?u.22404} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge 
valp: SurjVal p
valp
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha
SurjVal.v valp (e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (4 * e.toModel.a6) ≥ ofN (2 * q)
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q).
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha
SurjVal.v valp (e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha
SurjVal.v valp (e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (4 * e.toModel.a6) ≥ ofN (2 * q)rw [
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha
SurjVal.v valp (e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q)←
add_self_eq_mul_two: ∀ {R : Type ?u.22435} [inst : Semiring R] (a : R), a + a = 2 * a
add_self_eq_mul_two 
q: ℕ
q
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha
SurjVal.v valp (e.toModel.a3 * e.toModel.a3) ≥ ofN (q + q)]
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha
SurjVal.v valp (e.toModel.a3 * e.toModel.a3) ≥ ofN (q + q)
    
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha
SurjVal.v valp (e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (4 * e.toModel.a6) ≥ ofN (2 * q)exact (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.22505} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
valp: SurjVal p
valp 
h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 
h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3)
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q).
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (4 * e.toModel.a6) ≥ ofN (2 * q) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (4 * e.toModel.a6) ≥ ofN (2 * q)exact (
val_mul_ge_of_right_ge: ∀ {R : Type ?u.22621} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
valp: SurjVal p
valp 
h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)
h6)
Goals accomplished! 🐙

lemma 
v_b8_of_v_ai: ∀ {q : ℕ} {p : R} (valp : SurjVal p) (e : ValidModel R),
  SurjVal.v valp e.toModel.a1 ≥ ofN 1 →
    SurjVal.v valp e.toModel.a2 = ofN 1 →
      SurjVal.v valp e.toModel.a3 ≥ ofN q →
        SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1) →
          SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q) → SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)
v_b8_of_v_ai {
p: R
p : 
R: Type u
R} (
valp: SurjVal p
valp : 
SurjVal: {R : Type ?u.22648} → R → [inst : IntegralDomain R] → Type ?u.22648
SurjVal 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.22657) → [inst : IntegralDomain R] → Type ?u.22657
ValidModel 
R: Type u
R) (
h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1 : 
valp: SurjVal p
valp.
v: {R : Type ?u.22741} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a1: {R : Type ?u.22748} → [inst : IntegralDomain R] → Model R → R
a1 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.22752
1)
  (
h2: SurjVal.v valp e.toModel.a2 = ofN 1
h2 : 
valp: SurjVal p
valp.
v: {R : Type ?u.22759} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a2: {R : Type ?u.22766} → [inst : IntegralDomain R] → Model R → R
a2 = 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.22770
1) (
h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 : 
valp: SurjVal p
valp.
v: {R : Type ?u.22776} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a3: {R : Type ?u.22783} → [inst : IntegralDomain R] → Model R → R
a3 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
q: ?m.22726
q) (
h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)
h4 : 
valp: SurjVal p
valp.
v: {R : Type ?u.22791} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a4: {R : Type ?u.22798} → [inst : IntegralDomain R] → Model R → R
a4 ≥ 
ofN: ℕ → ℕ∪∞
ofN (
q: ?m.22726
q + 
1: ?m.22805
1))
  (
h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)
h6 : 
valp: SurjVal p
valp.
v: {R : Type ?u.22862} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a6: {R : Type ?u.22869} → [inst : IntegralDomain R] → Model R → R
a6 ≥ 
ofN: ℕ → ℕ∪∞
ofN (
2: ?m.22876
2 * 
q: ?m.22726
q)) : 
valp: SurjVal p
valp.
v: {R : Type ?u.22938} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
b8: {R : Type ?u.22945} → [inst : IntegralDomain R] → Model R → R
b8 ≥ 
ofN: ℕ → ℕ∪∞
ofN (
2: ?m.22958
2 * 
q: ?m.22726
q + 
1: ?m.22968
1) := by
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)simp only [
Model.b8: {R : Type ?u.23061} → [inst : IntegralDomain R] → Model R → R
Model.b8]
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp
    (e.toModel.a1 * e.toModel.a1 * e.toModel.a6 - e.toModel.a1 * e.toModel.a3 * e.toModel.a4 +           4 * e.toModel.a2 * e.toModel.a6 +         e.toModel.a2 * e.toModel.a3 * e.toModel.a3 -       e.toModel.a4 * e.toModel.a4) ≥   ofN (2 * q + 1)
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)rw [
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp
    (e.toModel.a1 * e.toModel.a1 * e.toModel.a6 - e.toModel.a1 * e.toModel.a3 * e.toModel.a4 +           4 * e.toModel.a2 * e.toModel.a6 +         e.toModel.a2 * e.toModel.a3 * e.toModel.a3 -       e.toModel.a4 * e.toModel.a4) ≥   ofN (2 * q + 1)
sub_eq_add_neg: ∀ {G : Type ?u.23313} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_neg,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp
    (e.toModel.a1 * e.toModel.a1 * e.toModel.a6 - e.toModel.a1 * e.toModel.a3 * e.toModel.a4 +           4 * e.toModel.a2 * e.toModel.a6 +         e.toModel.a2 * e.toModel.a3 * e.toModel.a3 +       -(e.toModel.a4 * e.toModel.a4)) ≥   ofN (2 * q + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp
    (e.toModel.a1 * e.toModel.a1 * e.toModel.a6 - e.toModel.a1 * e.toModel.a3 * e.toModel.a4 +           4 * e.toModel.a2 * e.toModel.a6 +         e.toModel.a2 * e.toModel.a3 * e.toModel.a3 -       e.toModel.a4 * e.toModel.a4) ≥   ofN (2 * q + 1)
sub_eq_add_neg: ∀ {G : Type ?u.23436} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_neg
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp
    (e.toModel.a1 * e.toModel.a1 * e.toModel.a6 + -(e.toModel.a1 * e.toModel.a3 * e.toModel.a4) +           4 * e.toModel.a2 * e.toModel.a6 +         e.toModel.a2 * e.toModel.a3 * e.toModel.a3 +       -(e.toModel.a4 * e.toModel.a4)) ≥   ofN (2 * q + 1)]
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp
    (e.toModel.a1 * e.toModel.a1 * e.toModel.a6 + -(e.toModel.a1 * e.toModel.a3 * e.toModel.a4) +           4 * e.toModel.a2 * e.toModel.a6 +         e.toModel.a2 * e.toModel.a3 * e.toModel.a3 +       -(e.toModel.a4 * e.toModel.a4)) ≥   ofN (2 * q + 1)
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)repeat 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp
    (e.toModel.a1 * e.toModel.a1 * e.toModel.a6 + -(e.toModel.a1 * e.toModel.a3 * e.toModel.a4) +           4 * e.toModel.a2 * e.toModel.a6 +         e.toModel.a2 * e.toModel.a3 * e.toModel.a3 +       -(e.toModel.a4 * e.toModel.a4)) ≥   ofN (2 * q + 1)apply 
val_add_ge_of_ge: ∀ {R : Type ?u.23596} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge 
valp: SurjVal p
valp
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.ha
SurjVal.v valp (e.toModel.a1 * e.toModel.a1 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hb
SurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hb
SurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1).
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.ha
SurjVal.v valp (e.toModel.a1 * e.toModel.a1 * e.toModel.a6) ≥ ofN (2 * q + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.ha
SurjVal.v valp (e.toModel.a1 * e.toModel.a1 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hb
SurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hb
SurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)rw [
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.ha
SurjVal.v valp (e.toModel.a1 * e.toModel.a1 * e.toModel.a6) ≥ ofN (2 * q + 1)
mul_assoc: ∀ {G : Type ?u.23621} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assoc,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.ha
SurjVal.v valp (e.toModel.a1 * (e.toModel.a1 * e.toModel.a6)) ≥ ofN (2 * q + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.ha
SurjVal.v valp (e.toModel.a1 * e.toModel.a1 * e.toModel.a6) ≥ ofN (2 * q + 1)
Nat.add_comm: ∀ (n m : ℕ), n + m = m + n
Nat.add_comm,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.ha
SurjVal.v valp (e.toModel.a1 * (e.toModel.a1 * e.toModel.a6)) ≥ ofN (1 + 2 * q) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.ha
SurjVal.v valp (e.toModel.a1 * e.toModel.a1 * e.toModel.a6) ≥ ofN (2 * q + 1)←
add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.ha
SurjVal.v valp (e.toModel.a1 * (e.toModel.a1 * e.toModel.a6)) ≥ ofN 1 + ofN (2 * q)]
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.ha
SurjVal.v valp (e.toModel.a1 * (e.toModel.a1 * e.toModel.a6)) ≥ ofN 1 + ofN (2 * q)
    
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.ha
SurjVal.v valp (e.toModel.a1 * e.toModel.a1 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hb
SurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hb
SurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)exact 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.23775} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
valp: SurjVal p
valp (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.23791} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
valp: SurjVal p
valp 
h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1 
h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)
h6)
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1).
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hb
SurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hb
SurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hb
SurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)rw [
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hb
SurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)
val_neg: ∀ {R : Type ?u.23812} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hb
SurjVal.v valp (e.toModel.a1 * e.toModel.a3 * e.toModel.a4) ≥ ofN (2 * q + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hb
SurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)
mul_assoc: ∀ {G : Type ?u.23830} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assoc,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hb
SurjVal.v valp (e.toModel.a1 * (e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hb
SurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)←
add_self_eq_mul_two: ∀ {R : Type ?u.23914} [inst : Semiring R] (a : R), a + a = 2 * a
add_self_eq_mul_two 
q: ℕ
q,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hb
SurjVal.v valp (e.toModel.a1 * (e.toModel.a3 * e.toModel.a4)) ≥ ofN (q + q + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hb
SurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)
Nat.add_assoc: ∀ (n m k : ℕ), n + m + k = n + (m + k)
Nat.add_assoc,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hb
SurjVal.v valp (e.toModel.a1 * (e.toModel.a3 * e.toModel.a4)) ≥ ofN (q + (q + 1)) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hb
SurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)←
add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hb
SurjVal.v valp (e.toModel.a1 * (e.toModel.a3 * e.toModel.a4)) ≥ ofN q + ofN (q + 1)]
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hb
SurjVal.v valp (e.toModel.a1 * (e.toModel.a3 * e.toModel.a4)) ≥ ofN q + ofN (q + 1)
    
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hb
SurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hb
SurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)exact 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.23990} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
valp: SurjVal p
valp (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.24000} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
valp: SurjVal p
valp 
h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 
h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)
h4)
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1).
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hb
SurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hb
SurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)rw [
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hb
SurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)
mul_assoc: ∀ {G : Type ?u.24011} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assoc,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hb
SurjVal.v valp (4 * (e.toModel.a2 * e.toModel.a6)) ≥ ofN (2 * q + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hb
SurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)
Nat.add_comm: ∀ (n m : ℕ), n + m = m + n
Nat.add_comm,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hb
SurjVal.v valp (4 * (e.toModel.a2 * e.toModel.a6)) ≥ ofN (1 + 2 * q) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hb
SurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)←
add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hb
SurjVal.v valp (4 * (e.toModel.a2 * e.toModel.a6)) ≥ ofN 1 + ofN (2 * q)]
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hb
SurjVal.v valp (4 * (e.toModel.a2 * e.toModel.a6)) ≥ ofN 1 + ofN (2 * q)
    
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hb
SurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)exact 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.24118} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
valp: SurjVal p
valp (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.24128} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
valp: SurjVal p
valp (
le_of_eq: ∀ {α : Type ?u.24139} [inst : Preorder α] {a b : α}, a = b → a ≤ b
le_of_eq 
h2: SurjVal.v valp e.toModel.a2 = ofN 1
h2.
symm: ∀ {α : Sort ?u.24146} {a b : α}, a = b → b = a
symm) 
h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)
h6)
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1).
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)rw [
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
mul_assoc: ∀ {G : Type ?u.24157} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assoc,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * (e.toModel.a3 * e.toModel.a3)) ≥ ofN (2 * q + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)←
add_self_eq_mul_two: ∀ {R : Type ?u.24241} [inst : Semiring R] (a : R), a + a = 2 * a
add_self_eq_mul_two 
q: ℕ
q,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * (e.toModel.a3 * e.toModel.a3)) ≥ ofN (q + q + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
Nat.add_comm: ∀ (n m : ℕ), n + m = m + n
Nat.add_comm,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * (e.toModel.a3 * e.toModel.a3)) ≥ ofN (1 + (q + q)) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)←
add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * (e.toModel.a3 * e.toModel.a3)) ≥ ofN 1 + ofN (q + q) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)←
add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * (e.toModel.a3 * e.toModel.a3)) ≥ ofN 1 + (ofN q + ofN q)]
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * (e.toModel.a3 * e.toModel.a3)) ≥ ofN 1 + (ofN q + ofN q)
    
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hb
SurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)exact 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.24277} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
valp: SurjVal p
valp (
le_of_eq: ∀ {α : Type ?u.24288} [inst : Preorder α] {a b : α}, a = b → a ≤ b
le_of_eq 
h2: SurjVal.v valp e.toModel.a2 = ofN 1
h2.
symm: ∀ {α : Sort ?u.24293} {a b : α}, a = b → b = a
symm) (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.24301} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
valp: SurjVal p
valp 
h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 
h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3)
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1).
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)rw [
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)
val_neg: ∀ {R : Type ?u.24312} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (e.toModel.a4 * e.toModel.a4) ≥ ofN (2 * q + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)←
add_self_eq_mul_two: ∀ {R : Type ?u.24327} [inst : Semiring R] (a : R), a + a = 2 * a
add_self_eq_mul_two 
q: ℕ
q,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (e.toModel.a4 * e.toModel.a4) ≥ ofN (q + q + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)
Nat.add_assoc: ∀ (n m k : ℕ), n + m + k = n + (m + k)
Nat.add_assoc,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (e.toModel.a4 * e.toModel.a4) ≥ ofN (q + (q + 1)) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)←
add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (e.toModel.a4 * e.toModel.a4) ≥ ofN q + ofN (q + 1)]
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (e.toModel.a4 * e.toModel.a4) ≥ ofN q + ofN (q + 1)
    
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hb
SurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)exact 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.24358} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
valp: SurjVal p
valp (
Enat.le_trans: ∀ {n m k : ℕ∪∞}, n ≤ m → m ≤ k → n ≤ k
Enat.le_trans (
le_succ: ∀ (n : ℕ∪∞), n ≤ succ n
le_succ (
ofN: ℕ → ℕ∪∞
ofN 
q: ℕ
q)) 
h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)
h4) 
h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)
h4
Goals accomplished! 🐙


lemma 
v_discr_of_v_ai: ∀ {q : ℕ} {p : R} (valp : SurjVal p) (e : ValidModel R),
  q > 1 →
    SurjVal.v valp e.toModel.a1 ≥ ofN 1 →
      SurjVal.v valp e.toModel.a2 = ofN 1 →
        SurjVal.v valp e.toModel.a3 ≥ ofN q →
          SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1) →
            SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q) → SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)
v_discr_of_v_ai {
p: R
p : 
R: Type u
R} (
valp: SurjVal p
valp : 
SurjVal: {R : Type ?u.24442} → R → [inst : IntegralDomain R] → Type ?u.24442
SurjVal 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.24431) → [inst : IntegralDomain R] → Type ?u.24431
ValidModel 
R: Type u
R) (
hq: q > 1
hq : 
q: ?m.24437
q > 
1: ?m.24453
1)
  (
h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1 : 
valp: SurjVal p
valp.
v: {R : Type ?u.24490} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a1: {R : Type ?u.24502} → [inst : IntegralDomain R] → Model R → R
a1 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.24508
1) (
h2: SurjVal.v valp e.toModel.a2 = ofN 1
h2 : 
valp: SurjVal p
valp.
v: {R : Type ?u.24526} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a2: {R : Type ?u.24533} → [inst : IntegralDomain R] → Model R → R
a2 = 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.24537
1) (
h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 : 
valp: SurjVal p
valp.
v: {R : Type ?u.24543} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a3: {R : Type ?u.24550} → [inst : IntegralDomain R] → Model R → R
a3 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
q: ?m.24437
q)
  (
h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)
h4 : 
valp: SurjVal p
valp.
v: {R : Type ?u.24558} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a4: {R : Type ?u.24565} → [inst : IntegralDomain R] → Model R → R
a4 ≥ 
ofN: ℕ → ℕ∪∞
ofN (
q: ?m.24437
q + 
1: ?m.24572
1)) (
h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)
h6 : 
valp: SurjVal p
valp.
v: {R : Type ?u.24629} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
a6: {R : Type ?u.24636} → [inst : IntegralDomain R] → Model R → R
a6 ≥ 
ofN: ℕ → ℕ∪∞
ofN (
2: ?m.24643
2 * 
q: ?m.24437
q)) :
  
valp: SurjVal p
valp.
v: {R : Type ?u.24705} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
discr: {R : Type ?u.24712} → [inst : IntegralDomain R] → Model R → R
discr ≥ 
ofN: ℕ → ℕ∪∞
ofN (
2: ?m.24725
2 * 
q: ?m.24437
q + 
3: ?m.24735
3) := by
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)have 
h2': ?m.24828
h2' := 
v_b2_of_v_a1_a2: ∀ {R : Type ?u.24829} [inst : IntegralDomain R] {p : R} (valp : SurjVal p) (e : ValidModel R),
  SurjVal.v valp e.toModel.a1 ≥ ofN 1 →
    SurjVal.v valp e.toModel.a2 = ofN 1 → SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1
v_b2_of_v_a1_a2 
valp: SurjVal p
valp 
e: ValidModel R
e 
h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1 
h2: SurjVal.v valp e.toModel.a2 = ofN 1
h2
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)have 
h4': ?m.24848
h4' := 
v_b4_of_v_a1_a3_a4: ∀ {R : Type ?u.24849} [inst : IntegralDomain R] {q : ℕ} {p : R} (valp : SurjVal p) (e : ValidModel R),
  SurjVal.v valp e.toModel.a1 ≥ ofN 1 →
    SurjVal.v valp e.toModel.a3 ≥ ofN q →
      SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1) → SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)
v_b4_of_v_a1_a3_a4 
valp: SurjVal p
valp 
e: ValidModel R
e 
h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1 
h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 
h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)
h4
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)have 
h6': ?m.24863
h6' := 
v_b6_of_v_a3_a6: ∀ {R : Type ?u.24864} [inst : IntegralDomain R] {q : ℕ} {p : R} (valp : SurjVal p) (e : ValidModel R),
  SurjVal.v valp e.toModel.a3 ≥ ofN q →
    SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q) → SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)
v_b6_of_v_a3_a6 
valp: SurjVal p
valp 
e: ValidModel R
e 
h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 
h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)
h6
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)have 
h8': ?m.24877
h8' := 
v_b8_of_v_ai: ∀ {R : Type ?u.24878} [inst : IntegralDomain R] {q : ℕ} {p : R} (valp : SurjVal p) (e : ValidModel R),
  SurjVal.v valp e.toModel.a1 ≥ ofN 1 →
    SurjVal.v valp e.toModel.a2 = ofN 1 →
      SurjVal.v valp e.toModel.a3 ≥ ofN q →
        SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1) →
          SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q) → SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)
v_b8_of_v_ai 
valp: SurjVal p
valp 
e: ValidModel R
e 
h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1 
h2: SurjVal.v valp e.toModel.a2 = ofN 1
h2 
h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 
h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)
h4 
h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)
h6
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)simp only [
Model.discr: {R : Type ?u.24897} → [inst : IntegralDomain R] → Model R → R
Model.discr]
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

SurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel - 8 * Model.b4 e.toModel ^ 3 -         27 * Model.b6 e.toModel * Model.b6 e.toModel +       9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥   ofN (2 * q + 3)
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)rw [
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

SurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel - 8 * Model.b4 e.toModel ^ 3 -         27 * Model.b6 e.toModel * Model.b6 e.toModel +       9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥   ofN (2 * q + 3)
sub_eq_add_neg: ∀ {G : Type ?u.25231} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_neg,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

SurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel - 8 * Model.b4 e.toModel ^ 3 +         -(27 * Model.b6 e.toModel * Model.b6 e.toModel) +       9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥   ofN (2 * q + 3) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

SurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel - 8 * Model.b4 e.toModel ^ 3 -         27 * Model.b6 e.toModel * Model.b6 e.toModel +       9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥   ofN (2 * q + 3)
sub_eq_add_neg: ∀ {G : Type ?u.25354} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_neg
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

SurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3) +         -(27 * Model.b6 e.toModel * Model.b6 e.toModel) +       9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥   ofN (2 * q + 3)]
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

SurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3) +         -(27 * Model.b6 e.toModel * Model.b6 e.toModel) +       9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥   ofN (2 * q + 3)
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)apply 
val_add_ge_of_ge: ∀ {R : Type ?u.25428} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge 
valp: SurjVal p
valp
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha
SurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3) +       -(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥   ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3).
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha
SurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3) +       -(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥   ofN (2 * q + 3) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha
SurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3) +       -(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥   ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)apply 
val_add_ge_of_ge: ∀ {R : Type ?u.25455} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge 
valp: SurjVal p
valp
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha
SurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3)) ≥   ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)
    
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha
SurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3) +       -(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥   ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3).
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha
SurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3)) ≥   ofN (2 * q + 3) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha
SurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3)) ≥   ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)apply 
val_add_ge_of_ge: ∀ {R : Type ?u.25473} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge 
valp: SurjVal p
valp
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)
      
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha
SurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3)) ≥   ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3).
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)rw [
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)←
neg_mul_left: ∀ {R : Type ?u.25489} [inst : Ring R] (a b : R), -(a * b) = -a * b
neg_mul_left,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (-(Model.b2 e.toModel * Model.b2 e.toModel) * Model.b8 e.toModel) ≥ ofN (2 * q + 3) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)←
neg_mul_left: ∀ {R : Type ?u.25579} [inst : Ring R] (a b : R), -(a * b) = -a * b
neg_mul_left,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (-(Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel)) ≥ ofN (2 * q + 3) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)
val_neg: ∀ {R : Type ?u.25617} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)(show 
3: ?m.25635
3 = 
1: ?m.25655
1 + 
1: ?m.25666
1 + 
1: ?m.25677
1 by 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)rfl
Goals accomplished! 🐙),
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + (1 + 1 + 1)) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)←
Nat.add_assoc: ∀ (n m k : ℕ), n + m + k = n + (m + k)
Nat.add_assoc,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + (1 + 1) + 1)
          
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)←
Nat.add_assoc: ∀ (n m k : ℕ), n + m + k = n + (m + k)
Nat.add_assoc,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 1 + 1 + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)
Nat.add_assoc: ∀ (n m k : ℕ), n + m + k = n + (m + k)
Nat.add_assoc,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 1 + (1 + 1)) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)
mul_comm: ∀ {G : Type ?u.25973} [inst : CommSemigroup G] (a b : G), a * b = b * a
mul_comm,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (Model.b8 e.toModel * (Model.b2 e.toModel * Model.b2 e.toModel)) ≥ ofN (2 * q + 1 + (1 + 1)) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)←
add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (Model.b8 e.toModel * (Model.b2 e.toModel * Model.b2 e.toModel)) ≥ ofN (2 * q + 1) + ofN (1 + 1)]
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (Model.b8 e.toModel * (Model.b2 e.toModel * Model.b2 e.toModel)) ≥ ofN (2 * q + 1) + ofN (1 + 1)
        
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.ha
SurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)exact 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.26050} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
valp: SurjVal p
valp 
h8': ?m.24877
h8' (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.26073} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
valp: SurjVal p
valp 
h2': ?m.24828
h2' 
h2': ?m.24828
h2')
Goals accomplished! 🐙
      
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha
SurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3)) ≥   ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3).
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)rw [
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)
val_neg: ∀ {R : Type ?u.26187} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (8 * Model.b4 e.toModel ^ 3) ≥ ofN (2 * q + 3) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)
pow_succ': ∀ {M : Type ?u.26202} [inst : Monoid M] (a : M) (n : ℕ), a ^ (n + 1) = a ^ n * a
pow_succ',
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (8 * (Model.b4 e.toModel ^ 2 * Model.b4 e.toModel)) ≥ ofN (2 * q + 3) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)
pow_succ': ∀ {M : Type ?u.26319} [inst : Monoid M] (a : M) (n : ℕ), a ^ (n + 1) = a ^ n * a
pow_succ',
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (8 * (Model.b4 e.toModel ^ 1 * Model.b4 e.toModel * Model.b4 e.toModel)) ≥ ofN (2 * q + 3) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)
pow_one: ∀ {M : Type ?u.26393} [inst : Monoid M] (a : M), a ^ 1 = a
pow_one,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (8 * (Model.b4 e.toModel * Model.b4 e.toModel * Model.b4 e.toModel)) ≥ ofN (2 * q + 3) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)←
add_self_eq_mul_two: ∀ {R : Type ?u.26465} [inst : Semiring R] (a : R), a + a = 2 * a
add_self_eq_mul_two,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (8 * (Model.b4 e.toModel * Model.b4 e.toModel * Model.b4 e.toModel)) ≥ ofN (q + q + 3)
          
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)(show 
q: ℕ
q + 
q: ℕ
q + 
3: ?m.26577
3 = 
q: ℕ
q + 
1: ?m.26600
1 + (
q: ℕ
q + 
1: ?m.26614
1) + 
1: ?m.26625
1 by 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (8 * (Model.b4 e.toModel * Model.b4 e.toModel * Model.b4 e.toModel)) ≥ ofN (q + q + 3)ring
Goals accomplished! 🐙),
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (8 * (Model.b4 e.toModel * Model.b4 e.toModel * Model.b4 e.toModel)) ≥ ofN (q + 1 + (q + 1) + 1) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)←
add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (8 * (Model.b4 e.toModel * Model.b4 e.toModel * Model.b4 e.toModel)) ≥ ofN (q + 1 + (q + 1)) + ofN 1 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)←
add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (8 * (Model.b4 e.toModel * Model.b4 e.toModel * Model.b4 e.toModel)) ≥ ofN (q + 1) + ofN (q + 1) + ofN 1]
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (8 * (Model.b4 e.toModel * Model.b4 e.toModel * Model.b4 e.toModel)) ≥ ofN (q + 1) + ofN (q + 1) + ofN 1
        
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hb
SurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)exact 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.26951} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
valp: SurjVal p
valp (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.26961} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
valp: SurjVal p
valp
          (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.26972} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
valp: SurjVal p
valp 
h4': ?m.24848
h4' 
h4': ?m.24848
h4') (
Enat.le_trans: ∀ {n m k : ℕ∪∞}, n ≤ m → m ≤ k → n ≤ k
Enat.le_trans ((
le_ofN: ∀ (m n : ℕ), m ≤ n ↔ ofN m ≤ ofN n
le_ofN 
_: ℕ
_ 
_: ℕ
_).
1: ∀ {a b : Prop}, (a ↔ b) → a → b
1 (
Nat.le_add_left: ∀ (n m : ℕ), n ≤ m + n
Nat.le_add_left 
1: ?m.26995
1 
q: ℕ
q)) 
h4': ?m.24848
h4'))
Goals accomplished! 🐙
    
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha
SurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3) +       -(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥   ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3).
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)rw [
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)
val_neg: ∀ {R : Type ?u.26997} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (27 * Model.b6 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)←
add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (27 * Model.b6 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q) + ofN 3 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)
mul_assoc: ∀ {G : Type ?u.27019} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assoc,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (27 * (Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q) + ofN 3 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)(show 
3: ?m.27146
3 = 
2: ?m.27163
2 + 
1: ?m.27173
1 by 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (27 * (Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q) + ofN 3rfl
Goals accomplished! 🐙)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (27 * (Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q) + ofN (2 + 1)]
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (27 * (Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q) + ofN (2 + 1)
      
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)apply 
val_mul_ge_of_right_ge: ∀ {R : Type ?u.27327} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
valp: SurjVal p
valp (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.27337} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
valp: SurjVal p
valp 
h6': ?m.24863
h6' (
Enat.le_trans: ∀ {n m k : ℕ∪∞}, n ≤ m → m ≤ k → n ≤ k
Enat.le_trans ((
le_ofN: ∀ (m n : ℕ), m ≤ n ↔ ofN m ≤ ofN n
le_ofN 
_: ℕ
_ 
_: ℕ
_).
1: ∀ {a b : Prop}, (a ↔ b) → a → b
1 
_: ?m.27353
_) 
h6': ?m.24863
h6'))
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

2 + 1 ≤ 2 * q
      
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)rw [
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

2 + 1 ≤ 2 * q←
add_self_eq_mul_two: ∀ {R : Type ?u.27356} [inst : Semiring R] (a : R), a + a = 2 * a
add_self_eq_mul_two 
q: ℕ
q
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

2 + 1 ≤ q + q]
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

2 + 1 ≤ q + q
      
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hb
SurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)exact 
Nat.add_le_add: ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → a + c ≤ b + d
Nat.add_le_add (
Nat.succ_le_of_lt: ∀ {n m : ℕ}, n < m → Nat.succ n ≤ m
Nat.succ_le_of_lt 
hq: q > 1
hq) (
Nat.le_of_lt: ∀ {n m : ℕ}, n < m → n ≤ m
Nat.le_of_lt 
hq: q > 1
hq)
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3).
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)rw [
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)(show 
3: ?m.27380
3 = 
1: ?m.27397
1 + (
1: ?m.27411
1 + 
1: ?m.27422
1) by 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)rfl
Goals accomplished! 🐙),
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + (1 + (1 + 1))) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)←
add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q) + ofN (1 + (1 + 1)) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)←
add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q) + (ofN 1 + ofN (1 + 1)) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)
mul_comm: ∀ {G : Type ?u.27706} [inst : CommSemigroup G] (a b : G), a * b = b * a
mul_comm,
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (Model.b6 e.toModel * (9 * Model.b2 e.toModel * Model.b4 e.toModel)) ≥   ofN (2 * q) + (ofN 1 + ofN (1 + 1)) 
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)
mul_assoc: ∀ {G : Type ?u.27742} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assoc 
9: ?m.27746
9
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (Model.b6 e.toModel * (9 * (Model.b2 e.toModel * Model.b4 e.toModel))) ≥   ofN (2 * q) + (ofN 1 + ofN (1 + 1))]
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (Model.b6 e.toModel * (9 * (Model.b2 e.toModel * Model.b4 e.toModel))) ≥   ofN (2 * q) + (ofN 1 + ofN (1 + 1))
    
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hb
SurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)exact 
val_mul_ge_of_both_ge: ∀ {R : Type ?u.27914} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
valp: SurjVal p
valp 
h6': ?m.24863
h6' (
val_mul_ge_of_right_ge: ∀ {R : Type ?u.27925} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge 
valp: SurjVal p
valp
      (
val_mul_ge_of_both_ge: ∀ {R : Type ?u.27935} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge 
valp: SurjVal p
valp 
h2': ?m.24828
h2' (
Enat.le_trans: ∀ {n m k : ℕ∪∞}, n ≤ m → m ≤ k → n ≤ k
Enat.le_trans ((
le_ofN: ∀ (m n : ℕ), m ≤ n ↔ ofN m ≤ ofN n
le_ofN 
_: ℕ
_ 
_: ℕ
_).
1: ∀ {a b : Prop}, (a ↔ b) → a → b
1
      (
Nat.add_le_add: ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → a + c ≤ b + d
Nat.add_le_add (
Nat.le_of_lt: ∀ {n m : ℕ}, n < m → n ≤ m
Nat.le_of_lt 
hq: q > 1
hq) (
le_of_eq: ∀ {α : Type ?u.27963} [inst : Preorder α] {a b : α}, a = b → a ≤ b
le_of_eq 
rfl: ∀ {α : Sort ?u.27980} {a : α}, a = a
rfl))) 
h4': ?m.24848
h4')))
Goals accomplished! 🐙

lemma 
small_char_div_12: ∀ {p : R}, p = 2 ∨ p = 3 → ∀ (valp : SurjVal p), SurjVal.v valp 12 ≥ ofN 1
small_char_div_12 {
p: R
p : 
R: Type u
R} (
hp: p = 2 ∨ p = 3
hp : 
p: R
p = 
2: ?m.28044
2 ∨ 
p: R
p = 
3: ?m.28122
3) (
valp: SurjVal p
valp : 
SurjVal: {R : Type ?u.28176} → R → [inst : IntegralDomain R] → Type ?u.28176
SurjVal 
p: R
p) : 
valp: SurjVal p
valp.
v: {R : Type ?u.28184} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
12: ?m.28192
12 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.28233
1 := by
  
R: Type u

inst: IntegralDomain R

p: R

hp: p = 2 ∨ p = 3

valp: SurjVal p

SurjVal.v valp 12 ≥ ofN 1cases 
hp: p = 2 ∨ p = 3
hp with
  
R: Type u

inst: IntegralDomain R

p: R

hp: p = 2 ∨ p = 3

valp: SurjVal p

SurjVal.v valp 12 ≥ ofN 1| 
inl: ∀ {a b : Prop}, a → a ∨ b
inl 
p2: ?m.28273
p2 =>
    
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inl
SurjVal.v valp 12 ≥ ofN 1rw [
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inl
SurjVal.v valp 12 ≥ ofN 1(show (
12: ?m.28307
12 : 
R: Type u
R) = 
2: ?m.28313
2 * 
6: ?m.28323
6 by 
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inl
SurjVal.v valp 12 ≥ ofN 1norm_num
Goals accomplished! 🐙)
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inl
SurjVal.v valp (2 * 6) ≥ ofN 1]
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inl
SurjVal.v valp (2 * 6) ≥ ofN 1
    
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inl
SurjVal.v valp 12 ≥ ofN 1apply 
val_mul_ge_of_left_ge: ∀ {R : Type ?u.28575} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_left_ge
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inl.ha
SurjVal.v valp 2 ≥ ofN 1
    
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inl
SurjVal.v valp 12 ≥ ofN 1rw [
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inl.ha
SurjVal.v valp 2 ≥ ofN 1←
p2: ?m.28273
p2
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inl.ha
SurjVal.v valp p ≥ ofN 1]
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inl.ha
SurjVal.v valp p ≥ ofN 1
    
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inl
SurjVal.v valp 12 ≥ ofN 1exact 
le_of_eq: ∀ {α : Type ?u.28619} [inst : Preorder α] {a b : α}, a = b → a ≤ b
le_of_eq (
valp: SurjVal p
valp.
v_uniformizer: ∀ {R : Type ?u.28630} {p : R} [inst : IntegralDomain R] (self : SurjVal p), SurjVal.v self p = ofN 1
v_uniformizer).
symm: ∀ {α : Sort ?u.28634} {a b : α}, a = b → b = a
symm
Goals accomplished! 🐙
  
R: Type u

inst: IntegralDomain R

p: R

hp: p = 2 ∨ p = 3

valp: SurjVal p

SurjVal.v valp 12 ≥ ofN 1| 
inr: ∀ {a b : Prop}, b → a ∨ b
inr 
p3: ?m.28274
p3 =>
    
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inr
SurjVal.v valp 12 ≥ ofN 1rw [
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inr
SurjVal.v valp 12 ≥ ofN 1(show (
12: ?m.28661
12 : 
R: Type u
R) = 
3: ?m.28667
3 * 
4: ?m.28677
4 by 
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inr
SurjVal.v valp 12 ≥ ofN 1norm_num
Goals accomplished! 🐙)
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inr
SurjVal.v valp (3 * 4) ≥ ofN 1]
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inr
SurjVal.v valp (3 * 4) ≥ ofN 1
    
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inr
SurjVal.v valp 12 ≥ ofN 1apply 
val_mul_ge_of_left_ge: ∀ {R : Type ?u.28826} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_left_ge
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inr.ha
SurjVal.v valp 3 ≥ ofN 1
    
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inr
SurjVal.v valp 12 ≥ ofN 1rw [
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inr.ha
SurjVal.v valp 3 ≥ ofN 1←
p3: ?m.28274
p3
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inr.ha
SurjVal.v valp p ≥ ofN 1]
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inr.ha
SurjVal.v valp p ≥ ofN 1
    
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inr
SurjVal.v valp 12 ≥ ofN 1exact 
le_of_eq: ∀ {α : Type ?u.28860} [inst : Preorder α] {a b : α}, a = b → a ≤ b
le_of_eq (
valp: SurjVal p
valp.
v_uniformizer: ∀ {R : Type ?u.28865} {p : R} [inst : IntegralDomain R] (self : SurjVal p), SurjVal.v self p = ofN 1
v_uniformizer).
symm: ∀ {α : Sort ?u.28869} {a b : α}, a = b → b = a
symm
Goals accomplished! 🐙


lemma 
v_rst_b2_of_small_char: ∀ {p : R} (valp : SurjVal p) (e : ValidModel R) (r s t : R),
  SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1 →
    SurjVal.v valp 12 ≥ ofN 1 → SurjVal.v valp (Model.b2 (rst_iso r s t e).toModel) ≥ ofN 1
v_rst_b2_of_small_char {
p: R
p : 
R: Type u
R} (
valp: SurjVal p
valp : 
SurjVal: {R : Type ?u.28902} → R → [inst : IntegralDomain R] → Type ?u.28902
SurjVal 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.28911) → [inst : IntegralDomain R] → Type ?u.28911
ValidModel 
R: Type u
R) (
r: R
r 
s: R
s 
t: R
t : 
R: Type u
R)
  (
h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1
h_b2 : 
valp: SurjVal p
valp.
v: {R : Type ?u.28922} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
e: ValidModel R
e.
b2: {R : Type ?u.28934} → [inst : IntegralDomain R] → Model R → R
b2 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.28943
1) (
h_p: SurjVal.v valp 12 ≥ ofN 1
h_p : 
valp: SurjVal p
valp.
v: {R : Type ?u.28961} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
12: ?m.28966
12 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.29029
1) : 
valp: SurjVal p
valp.
v: {R : Type ?u.29036} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v (
rst_iso: {R : Type ?u.29040} → [inst : IntegralDomain R] → R → R → R → ValidModel R → ValidModel R
rst_iso 
r: R
r 
s: R
s 
t: R
t 
e: ValidModel R
e).
b2: {R : Type ?u.29046} → [inst : IntegralDomain R] → Model R → R
b2 ≥ 
ofN: ℕ → ℕ∪∞
ofN 
1: ?m.29053
1 := by
  
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

r, s, t: R

h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h_p: SurjVal.v valp 12 ≥ ofN 1

SurjVal.v valp (Model.b2 (rst_iso r s t e).toModel) ≥ ofN 1simp only [
rst_iso: {R : Type ?u.29083} → [inst : IntegralDomain R] → R → R → R → ValidModel R → ValidModel R
rst_iso]
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

r, s, t: R

h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h_p: SurjVal.v valp 12 ≥ ofN 1

SurjVal.v valp (Model.b2 (Model.rst_iso r s t e.toModel)) ≥ ofN 1
  
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

r, s, t: R

h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h_p: SurjVal.v valp 12 ≥ ofN 1

SurjVal.v valp (Model.b2 (rst_iso r s t e).toModel) ≥ ofN 1rw [
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

r, s, t: R

h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h_p: SurjVal.v valp 12 ≥ ofN 1

SurjVal.v valp (Model.b2 (Model.rst_iso r s t e.toModel)) ≥ ofN 1
Model.rst_b2: ∀ {R : Type ?u.29177} [inst : IntegralDomain R] (r s t : R) (e : Model R),
  Model.b2 (Model.rst_iso r s t e) = Model.b2 e + 12 * r
Model.rst_b2
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

r, s, t: R

h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h_p: SurjVal.v valp 12 ≥ ofN 1

SurjVal.v valp (Model.b2 e.toModel + 12 * r) ≥ ofN 1]
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

r, s, t: R

h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h_p: SurjVal.v valp 12 ≥ ofN 1

SurjVal.v valp (Model.b2 e.toModel + 12 * r) ≥ ofN 1
  
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

r, s, t: R

h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h_p: SurjVal.v valp 12 ≥ ofN 1

SurjVal.v valp (Model.b2 (rst_iso r s t e).toModel) ≥ ofN 1apply 
val_add_ge_of_ge: ∀ {R : Type ?u.29205} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge 
valp: SurjVal p
valp 
h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1
h_b2
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

r, s, t: R

h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h_p: SurjVal.v valp 12 ≥ ofN 1

SurjVal.v valp (12 * r) ≥ ofN 1
  
R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

r, s, t: R

h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h_p: SurjVal.v valp 12 ≥ ofN 1

SurjVal.v valp (Model.b2 (rst_iso r s t e).toModel) ≥ ofN 1exact 
val_mul_ge_of_left_ge: ∀ {R : Type ?u.29233} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_left_ge 
valp: SurjVal p
valp 
h_p: SurjVal.v valp 12 ≥ ofN 1
h_p
Goals accomplished! 🐙

section cubic

def 
Δcubic: {R : Type u} → [inst : IntegralDomain R] → R × R × R → R
Δcubic (
c: R × R × R
c : 
R: Type u
R × 
R: Type u
R × 
R: Type u
R) : 
R: Type u
R := 
18: ?m.29296
18 * 
c: R × R × R
c.
1: {α : Type ?u.29306} → {β : Type ?u.29305} → α × β → α
1 * 
c: R × R × R
c.
2: {α : Type ?u.29312} → {β : Type ?u.29311} → α × β → β
2.
1: {α : Type ?u.29316} → {β : Type ?u.29315} → α × β → α
1 * 
c: R × R × R
c.
2: {α : Type ?u.29320} → {β : Type ?u.29319} → α × β → β
2.
2: {α : Type ?u.29324} → {β : Type ?u.29323} → α × β → β
2 - 
4: ?m.29334
4 * 
c: R × R × R
c.
1: {α : Type ?u.29347} → {β : Type ?u.29346} → α × β → α
1 ^ 
3: ?m.29351
3 * 
c: R × R × R
c.
2: {α : Type ?u.29361} → {β : Type ?u.29360} → α × β → β
2.
2: {α : Type ?u.29365} → {β : Type ?u.29364} → α × β → β
2 + 
c: R × R × R
c.
1: {α : Type ?u.29375} → {β : Type ?u.29374} → α × β → α
1 ^ 
2: ?m.29379
2 * 
c: R × R × R
c.
2: {α : Type ?u.29392} → {β : Type ?u.29391} → α × β → β
2.
1: {α : Type ?u.29396} → {β : Type ?u.29395} → α × β → α
1 ^ 
2: ?m.29400
2 - 
4: ?m.29413
4 * 
c: R × R × R
c.
2: {α : Type ?u.29426} → {β : Type ?u.29425} → α × β → β
2.
1: {α : Type ?u.29430} → {β : Type ?u.29429} → α × β → α
1 ^ 
3: ?m.29434
3 - 
27: ?m.29447
27 * 
c: R × R × R
c.
2: {α : Type ?u.29460} → {β : Type ?u.29459} → α × β → β
2.
2: {α : Type ?u.29464} → {β : Type ?u.29463} → α × β → β
2 ^ 
2: ?m.29468
2

def 
model_to_cubic: {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic {
p: R
p : 
R: Type u
R} (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.32385} → R → [inst : IntegralDomain R] → Type ?u.32385
EnatValRing 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.32394) → [inst : IntegralDomain R] → Type ?u.32394
ValidModel 
R: Type u
R) : 
R: Type u
R × 
R: Type u
R × 
R: Type u
R := (
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.32412} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: ValidModel R
e.
a2: {R : Type ?u.32430} → [inst : IntegralDomain R] → Model R → R
a2 
1: ?m.32436
1, 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.32450} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: ValidModel R
e.
a4: {R : Type ?u.32463} → [inst : IntegralDomain R] → Model R → R
a4 
2: ?m.32467
2, 
evr: EnatValRing p
evr.
sub_val: {R : Type ?u.32472} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val 
e: ValidModel R
e.
a6: {R : Type ?u.32485} → [inst : IntegralDomain R] → Model R → R
a6 
3: ?m.32489
3)

def 
cubic_has_dinstinct_roots: {p : R} → EnatValRing p → ValidModel R → Prop
cubic_has_dinstinct_roots {
p: R
p : 
R: Type u
R} (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.32564} → R → [inst : IntegralDomain R] → Type ?u.32564
EnatValRing 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.32573) → [inst : IntegralDomain R] → Type ?u.32573
ValidModel 
R: Type u
R) : 
Prop: Type
Prop := 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.32582} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.32589} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v (
Δcubic: {R : Type ?u.32596} → [inst : IntegralDomain R] → R × R × R → R
Δcubic (
model_to_cubic: {R : Type ?u.32599} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic 
evr: EnatValRing p
evr 
e: ValidModel R
e)) = 
0: ?m.32608
0

def 
δmultiplicity: R × R × R → R
δmultiplicity (
c: R × R × R
c : 
R: Type u
R × 
R: Type u
R × 
R: Type u
R) : 
R: Type u
R := 
3: ?m.32888
3 * 
c: R × R × R
c.
2: {α : Type ?u.32898} → {β : Type ?u.32897} → α × β → β
2.
1: {α : Type ?u.32904} → {β : Type ?u.32903} → α × β → α
1 - 
c: R × R × R
c.
1: {α : Type ?u.32911} → {β : Type ?u.32910} → α × β → α
1 ^ 
2: ?m.32915
2

def 
cubic_has_double_root: {p : R} → EnatValRing p → ValidModel R → Prop
cubic_has_double_root {
p: R
p : 
R: Type u
R} (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.33482} → R → [inst : IntegralDomain R] → Type ?u.33482
EnatValRing 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.33491) → [inst : IntegralDomain R] → Type ?u.33491
ValidModel 
R: Type u
R) : 
Prop: Type
Prop := 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.33500} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.33507} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v (
Δcubic: {R : Type ?u.33514} → [inst : IntegralDomain R] → R × R × R → R
Δcubic (
model_to_cubic: {R : Type ?u.33517} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic 
evr: EnatValRing p
evr 
e: ValidModel R
e)) > 
0: ?m.33526
0 ∧ 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.33780} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.33784} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v (
δmultiplicity: {R : Type ?u.33788} → [inst : IntegralDomain R] → R × R × R → R
δmultiplicity (
model_to_cubic: {R : Type ?u.33791} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic 
evr: EnatValRing p
evr 
e: ValidModel R
e)) = 
0: ?m.33796
0

def 
cubic_has_triple_root: {p : R} → EnatValRing p → ValidModel R → Prop
cubic_has_triple_root {
p: R
p : 
R: Type u
R} (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.33830} → R → [inst : IntegralDomain R] → Type ?u.33830
EnatValRing 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.33839) → [inst : IntegralDomain R] → Type ?u.33839
ValidModel 
R: Type u
R) : 
Prop: Type
Prop := 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.33848} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.33855} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v (
Δcubic: {R : Type ?u.33862} → [inst : IntegralDomain R] → R × R × R → R
Δcubic (
model_to_cubic: {R : Type ?u.33865} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic 
evr: EnatValRing p
evr 
e: ValidModel R
e)) > 
0: ?m.33874
0 ∧ 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.34128} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.34132} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v (
δmultiplicity: {R : Type ?u.34136} → [inst : IntegralDomain R] → R × R × R → R
δmultiplicity (
model_to_cubic: {R : Type ?u.34139} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic 
evr: EnatValRing p
evr 
e: ValidModel R
e)) > 
0: ?m.34144
0

def 
move_cubic_double_root_to_origin_iso: {p : R} → EnatValRing p → ValidModel R → ValidModel R
move_cubic_double_root_to_origin_iso {
p: R
p : 
R: Type u
R} (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.34179} → R → [inst : IntegralDomain R] → Type ?u.34179
EnatValRing 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.34188) → [inst : IntegralDomain R] → Type ?u.34188
ValidModel 
R: Type u
R) : 
ValidModel: (R : Type ?u.34192) → [inst : IntegralDomain R] → Type ?u.34192
ValidModel 
R: Type u
R :=
  let (
a2p: R
a2p, 
a4p2: R
a4p2, 
a6p3: R
a6p3
Warning: unused variable `a6p3` [linter.unusedVariables]) := 
model_to_cubic: {R : Type ?u.34198} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic 
evr: EnatValRing p
evr 
e: ValidModel R
e
  
rst_iso: {R : Type ?u.34255} → [inst : IntegralDomain R] → R → R → R → ValidModel R → ValidModel R
rst_iso (
p: R
p * (
evr: EnatValRing p
evr.
norm_repr: {R : Type ?u.34261} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → R → R
norm_repr (if 
evr: EnatValRing p
evr.
residue_char: {R : Type ?u.34266} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → ℕ
residue_char = 
2: ?m.34271
2 then 
a4p2: R
a4p2 else 
a2p: R
a2p * 
a4p2: R
a4p2))) 
0: ?m.34474
0 
0: ?m.34529
0 
e: ValidModel R
e

def 
cubic_double_root_is_zero: {p : R} → EnatValRing p → ValidModel R → Prop
cubic_double_root_is_zero {
p: R
p : 
R: Type u
R} (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.34865} → R → [inst : IntegralDomain R] → Type ?u.34865
EnatValRing 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.34874) → [inst : IntegralDomain R] → Type ?u.34874
ValidModel 
R: Type u
R) : 
Prop: Type
Prop :=
  let (
a2p: R
a2p, 
a4p2: R
a4p2, 
a6p3: R
a6p3) := 
model_to_cubic: {R : Type ?u.34882} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic 
evr: EnatValRing p
evr 
e: ValidModel R
e
  
evr: EnatValRing p
evr.
valtn: {R : Type ?u.34939} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.34943} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
a2p: R
a2p = 
0: ?m.34951
0 ∧ 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.35201} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.35205} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
a4p2: R
a4p2 > 
0: ?m.35210
0 ∧ 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.35231} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.35235} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
a6p3: R
a6p3 > 
0: ?m.35240
0

lemma 
move_cubic_double_root_to_origin: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : ValidModel R),
  cubic_has_double_root evr e → cubic_double_root_is_zero evr (move_cubic_double_root_to_origin_iso evr e)
move_cubic_double_root_to_origin
Warning: declaration uses 'sorry' {
p: R
p : 
R: Type u
R} (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.35366} → R → [inst : IntegralDomain R] → Type ?u.35366
EnatValRing 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.35375) → [inst : IntegralDomain R] → Type ?u.35375
ValidModel 
R: Type u
R) : 
cubic_has_double_root: {R : Type ?u.35380} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → Prop
cubic_has_double_root 
evr: EnatValRing p
evr 
e: ValidModel R
e → 
cubic_double_root_is_zero: {R : Type ?u.35393} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → Prop
cubic_double_root_is_zero 
evr: EnatValRing p
evr (
move_cubic_double_root_to_origin_iso: {R : Type ?u.35400} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → ValidModel R
move_cubic_double_root_to_origin_iso 
evr: EnatValRing p
evr 
e: ValidModel R
e) := 
sorry: ?m.35413
sorry

def 
move_cubic_triple_root_to_origin_iso: {p : R} → EnatValRing p → ValidModel R → ValidModel R
move_cubic_triple_root_to_origin_iso {
p: R
p : 
R: Type u
R} (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.35435} → R → [inst : IntegralDomain R] → Type ?u.35435
EnatValRing 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.35444) → [inst : IntegralDomain R] → Type ?u.35444
ValidModel 
R: Type u
R) : 
ValidModel: (R : Type ?u.35448) → [inst : IntegralDomain R] → Type ?u.35448
ValidModel 
R: Type u
R :=
  let (
a2p: R
a2p, 
a4p2: R
a4p2
Warning: unused variable `a4p2` [linter.unusedVariables], 
a6p3: R
a6p3) := 
model_to_cubic: {R : Type ?u.35454} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic 
evr: EnatValRing p
evr 
e: ValidModel R
e
  
rst_iso: {R : Type ?u.35511} → [inst : IntegralDomain R] → R → R → R → ValidModel R → ValidModel R
rst_iso (
p: R
p * (
evr: EnatValRing p
evr.
norm_repr: {R : Type ?u.35517} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → R → R
norm_repr (if 
evr: EnatValRing p
evr.
residue_char: {R : Type ?u.35522} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → ℕ
residue_char = 
2: ?m.35527
2 then -
a2p: R
a2p else -
a6p3: R
a6p3))) 
0: ?m.35707
0 
0: ?m.35762
0 
e: ValidModel R
e

def 
cubic_triple_root_is_zero: {p : R} → EnatValRing p → ValidModel R → Prop
cubic_triple_root_is_zero {
p: R
p : 
R: Type u
R} (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.36076} → R → [inst : IntegralDomain R] → Type ?u.36076
EnatValRing 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.36085) → [inst : IntegralDomain R] → Type ?u.36085
ValidModel 
R: Type u
R) : 
Prop: Type
Prop :=
  let (
a2p: R
a2p, 
a4p2: R
a4p2, 
a6p3: R
a6p3) := 
model_to_cubic: {R : Type ?u.36093} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic 
evr: EnatValRing p
evr 
e: ValidModel R
e
  
evr: EnatValRing p
evr.
valtn: {R : Type ?u.36150} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.36154} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
a2p: R
a2p > 
0: ?m.36162
0 ∧ 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.36416} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.36420} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
a4p2: R
a4p2 > 
0: ?m.36425
0 ∧ 
evr: EnatValRing p
evr.
valtn: {R : Type ?u.36443} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.
v: {R : Type ?u.36447} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 
a6p3: R
a6p3 > 
0: ?m.36452
0

lemma 
move_cubic_triple_root_to_origin: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : ValidModel R),
  cubic_has_triple_root evr e → cubic_triple_root_is_zero evr (move_cubic_triple_root_to_origin_iso evr e)
move_cubic_triple_root_to_origin
Warning: declaration uses 'sorry' {
p: R
p : 
R: Type u
R} (
evr: EnatValRing p
evr : 
EnatValRing: {R : Type ?u.36578} → R → [inst : IntegralDomain R] → Type ?u.36578
EnatValRing 
p: R
p) (
e: ValidModel R
e : 
ValidModel: (R : Type ?u.36587) → [inst : IntegralDomain R] → Type ?u.36587
ValidModel 
R: Type u
R) : 
cubic_has_triple_root: {R : Type ?u.36592} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → Prop
cubic_has_triple_root 
evr: EnatValRing p
evr 
e: ValidModel R
e → 
cubic_triple_root_is_zero: {R : Type ?u.36605} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → Prop
cubic_triple_root_is_zero 
evr: EnatValRing p
evr (
move_cubic_triple_root_to_origin_iso: {R : Type ?u.36612} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → ValidModel R
move_cubic_triple_root_to_origin_iso 
evr: EnatValRing p
evr 
e: ValidModel R
e) := 
sorry: ?m.36625
sorry

end cubic


end ValidModel